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Created by Mario Carneiro

Intuitionistic Logic Proof Explorer

Intuitionistic Logic (Wikipedia [accessed 19-Jul-2015], Stanford Encyclopedia of Philosophy [accessed 19-Jul-2015]) can be thought of as a constructive logic in which we must build and exhibit concrete examples of objects before we can accept their existence. Unproved statements in intuitionistic logic are not given an intermediate truth value, instead, they remain of unknown truth value until they are either proved or disproved. Intuitionist logic can also be thought of as a weakening of classical logic such that the law of excluded middle (LEM), (φ ¬ φ), doesn't always hold. Specifically, it holds if we have a proof for φ or we have a proof for ¬ φ, but it doesn't necessarily hold if we don't have a proof of either one. There is also no rule for double negation elimination. Brouwer observed in 1908 that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections.

 Contents of this page Overview of intuitionistic logic Overview of this work The axioms Some theorems How to intuitionize classical proofs Metamath Proof Explorer cross reference Bibliography Related pages Table of Contents and Theorem List Most Recent Proofs (this mirror) (latest) Bibliographic Cross-Reference Definition List ASCII Equivalents for Text-Only Browsers Metamath database iset.mm (ASCII file) External links GitHub repository [accessed 06-Jan-2018]

Overview of intuitionistic logic

(Placeholder for future use)

Overview of this work

(By Gérard Lang, 7-May-2018)

Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a development of "set.mm" whose eventual aim is to show how many of the theorems of set theory and mathematics that can be derived from classical first order logic can also be derived from a weaker system called "intuitionistic logic." To achieve this task, iset.mm adds (or substitutes) intuitionistic axioms for a number of the classical logical axioms of set.mm.

Among these new axioms, the first six (ax-ia1, ax-ia2, ax-ia3, ax-io, ax-in1, and ax-in2), when added to ax-1, ax-2, and ax-mp, allow for the development of intuitionistic propositional logic. We omit the classical axiom ((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑)) (which is ax-3 in set.mm). Each of our new axioms is a theorem of classical propositional logic, but ax-3 cannot be derived from them. Similarly, other basic classical theorems, like the third middle excluded or the equivalence of a proposition with its double negation, cannot be derived in intuitionistic propositional calculus. Glivenko showed that a proposition φ is a theorem of classical propositional calculus if and only if ¬¬φ is a theorem of intuitionistic propositional calculus.

The next 4 new axioms (ax-ial, ax-i5r, ax-ie1, and ax-ie2) together with the set.mm axioms ax-4, ax-5, ax-7, and ax-gen do not mention equality or distinct variables.

The ax-i9 axiom is just a slight variation of the classical ax-9. The classical axiom ax-12 is strengthened into first ax-i12 and then ax-bndl (two results which would be fairly readily equivalent to ax-12 classically but which do not follow from ax-12, at least not in an obvious way, in intuitionistic logic). The substitution of ax-i9, ax-i12, and ax-bndl for ax-9 and ax-12 and the inclusion of ax-8, ax-10, ax-11, ax-13, ax-14, and ax-17 allow for the development of the intuitionistic predicate calculus.

Each of the new axioms is a theorem of classical first order logic with equality. But some axioms of classical first order logic with equality, like ax-3, cannot be derived in the intuitionistic predicate calculus.

One of the major interests of the intuitionistic predicate calculus is that its use can be considered as a realization of the program of the constructivist philosophical view of mathematics.

The axioms

As with the classical axioms we have propositional logic and predicate logic.

The axioms of intuitionistic propositional logic consist of some of the axioms from classical propositional logic, plus additional axioms for the operation of the 'and', 'or' and 'not' connectives.

 Axiom Simp ax-1 ⊢ (φ → (ψ → φ)) Axiom Frege ax-2 ⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ))) Rule of Modus Ponens ax-mp &     ⇒ Left 'and' elimination ax-ia1 Right 'and' elimination ax-ia2 ⊢ ((φ ∧ ψ) → ψ) 'And' introduction ax-ia3 ⊢ (φ → (ψ → (φ ∧ ψ))) Definition of 'or' ax-io ⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ))) 'Not' introduction ax-in1 ⊢ ((φ → ¬ φ) → ¬ φ) 'Not' elimination ax-in2 ⊢ (¬ φ → (φ → ψ))

Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of implication and 'not'. In particular, φψ is not equivalent to ¬ φψ, nor is φψ equivalent to ¬ φψ, nor is φψ equivalent to ¬ (φ → ¬ ψ).

The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if φ implies a contradiction (such as its own negation), then one can conclude ¬ φ. By contrast, assuming ¬ φ and then deriving a contradiction only serves to prove ¬ ¬ φ, which in intuitionistic logic is not the same as φ.

The biconditional can be defined as the conjunction of two implications, as in dfbi2 and df-bi.

Predicate logic adds set variables (individual variables) and the ability to quantify them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃ cannot be defined in terms of ∀. As in classical logic, we also add = for equality (which is key to how we handle substitution in metamath) and ∈ (which for current purposes can just be thought of as an arbitrary predicate, but which will later come to mean set membership).

Our axioms are based on the classical set.mm predicate logic axioms, but adapted for intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing some aspects of how we handle negations.

 Axiom of Specialization ax-4 ⊢ (∀xφ → φ) Axiom of Quantified Implication ax-5 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) The converse of ax-5o ax-i5r ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ)) Axiom of Quantifier Commutation ax-7 ⊢ (∀x∀yφ → ∀y∀xφ) Rule of Generalization ax-gen ⊢ φ   =>    ⊢ ∀xφ x is bound in ∀xφ ax-ial ⊢ (∀xφ → ∀x∀xφ) x is bound in ∃xφ ax-ie1 ⊢ (∃xφ → ∀x∃xφ) Define existential quantification ax-ie2 ⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) Axiom of Equality ax-8 ⊢ (x = y → (x = z → y = z)) Axiom of Existence ax-i9 ⊢ ∃x x = y Axiom of Quantifier Substitution ax-10 ⊢ (∀x x = y → ∀y y = x) Axiom of Variable Substitution ax-11 ⊢ (x = y → (∀yφ → ∀x(x = y → φ))) Axiom of Quantifier Introduction ax-i12 ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) Axiom of Bundling ax-bndl ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y))) Left Membership Equality ax-13 ⊢ (x = y → (x ∈ z → y ∈ z)) Right Membership Equality ax-14 ⊢ (x = y → (z ∈ x → z ∈ y)) Distinctness ax-17 ⊢ (φ → ∀xφ), where x does not occur in φ

Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. These axioms were developed in response to Russell's Paradox, a discovery that revolutionized the foundations of mathematics and logic.

In the IZF axioms that follow, all set variables are assumed to be distinct. If you click on their links you will see the explicit distinct variable conditions.

 Axiom of Extensionality ax-ext ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y) Axiom of Collection ax-coll ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ) Axiom of Separation ax-sep ⊢ ∃y∀x(x ∈ y ↔ (x ∈ z ∧ φ)) Axiom of Power Sets ax-pow ⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y) Axiom of Pairing ax-pr ⊢ ∃z∀w((w = x ∨ w = y) → w ∈ z) Axiom of Union ax-un ⊢ ∃y∀z(∃w(z ∈ w ∧ w ∈ x) → z ∈ y) Axiom of Set Induction ax-setind ⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ) Axiom of Infinity ax-iinf ⊢ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))

We develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in metamath because the Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas.

A Theorem Sampler

From a psychological point of view, learning constructive mathematics is agonizing, for it requires one to first unlearn certain deeply ingrained intuitions and habits acquired during classical mathematical training.
—Andrej Bauer

Listed here are some examples of starting points for your journey through the maze. Some are stated just as they would be in a non-constructive context; others are here to highlight areas which look different constructively. You should study some simple proofs from propositional calculus until you get the hang of it. Then try some predicate calculus and finally set theory.

The Theorem List shows the complete set of theorems in the database. You may also find the Bibliographic Cross-Reference useful.

 Propositional calculus Law of identity Praeclarum theorema Contraposition introduction Double negation introduction Triple negation Definition of exclusive or Negation and the false constant Predicate calculus Existential and universal quantifier swap Existentially quantified implication x = x Definition of proper substitution Double existential uniqueness Set theory Commutative law for union A basic relationship between class and wff variables Two ways of saying "is a set" The ZF axiom of foundation implies excluded middle Russell's paradox Ordinal trichotomy implies excluded middle Mathematical (finite) induction Peano's postulates for arithmetic: 1 2 3 4 5 Two natural numbers are either equal or not equal (a special case of the law of the excluded middle which we can prove). A natural number is either zero or a successor The axiom of choice implies excluded middle Burali-Forti paradox Transfinite induction Closure law for ordinal addition Real and complex numbers Properties of apartness: 1 2 3 4

How to intuitionize classical proofs

For the student or master of classical mathematics, constructive mathematics can be baffling. One can get over some of the intial hurdles of understanding how constructive mathematics works and why it might be interesting by reading [Bauer] but that work does little to explain in concrete terms how to write proofs in intuitionistic logic. Fortunately, metamath helps with one of the biggest hurdles: noticing when one is even using the law of the excluded middle or the axiom of choice. But suppose you have a classical proof from the Metamath Proof Explorer and it fails to verify when you copy it over to the Intuitionistic Logic Explorer. What then? Here are some rules of thumb in converting classical proofs to intuitionistic ones.

• If you see case elimination ( pm2.61 or its variants) you'll probably end up with two theorems for the two cases. In particular, if the cases were and you probably just care about the case. On the other hand, if the proposition being eliminated is decidable (for example due to nndceq, zdceq, zdcle, zdclt, eluzdc, or fzdcel), then case elimination will work using theorems such as df-dc and mpjaodan.
• Non-empty almost always needs to be changed to inhabited (those terms are defined at n0rf).
• If the original proof relied on propositional/predicate logic which isn't a theorem of intuitionistic logic, maybe there is a way of expressing the logic more directly. This is perhaps the hardest one to put in cookbook form: you might need to try some things and see if anything works.
• If the original proof relied on df-ex so that it could prove a theorem for and then get for free (or vice versa), instead go look at the original proof and try to come up the analogues to each step for the other quantifier (for example, cbvrexcsf, sbcrext, rexxpf). Similarly, if you have a theorem for and are trying to prove the corresponding theorem for you'll probably need to use analogous steps rather than negation (examples: leaddsub, ltsub1, ltsub2).
• If you are dealing with a definition, try to find the best constructive definition from the literature ([HoTT] book, Stanford Encyclopedia of Philosophy, [Bauer], etc). Once you pick a definition, that'll affect the proofs which rely on that definition.
• If there is case elimination on whether variables are distinct, most of the time you just need the variant with distinct variables. Sometimes you can then remove the constraint with a temporary variable (e.g. the various sbco2* variants, nfralya, r19.3rm).
• Sometimes one direction of a biconditional holds, or subset holds instead of equality. You might be able to keep the easy direction and worry about the other one later.
• If there is case elimination sometimes only one of the two cases is possible. For example, in mosubopt the rest of the formula being proved constrains which case matters.
• If you need an additional condition (for example, because the original proof used case elimination) and you are proving a biconditional, consider whether both sides of the biconditional imply the condition. If so, you'll be able to prove the biconditional with that condition as an antecedent, and then use pm5.21nii or one of its variants to remove the antecedent (example: elxp4).
• If your proof relies on dveeq2 try dveeq2or and likewise for the other things downstream of ax-i12 or ax-bndl.
• If you have a disjunction, be reluctant to turn it into an implication using ord and the like. Instead, show that each disjunct implies what you are trying to prove and use jaoi to join those two implications into something which can hook up to the disjunction.
• Disjunctive syllogism holds in intuitionistic logic and we state it a few ways (for example orel1 and ecased) but we don't have a wide variety of convenience theorems. Unless we add those, you'll use ord or something similar followed by mpd or something similar. This may add a few steps but they are straightforward ones.
• If your proof is doing tricky things perhaps in the interest of shortness, try just expanding the definitions and applying logic in a straightforward way. See if this gets you a working (although perhaps longer) proof.
• If your proof relies on a biconditional in set.mm which isn't in iset.mm, see if one direction is in iset.mm and see which direction your proof is using. For example 19.35-1 or exnalim.
• If you are doing things with inhabited classes (beyond just applying existing inhabited class theorems), you may be able to dig up some predicate logic you haven't used in a while (e.g. raaan).
• Consider the possibility of giving up. Some things just won't have intuitionistic proofs. The more it looks like excluded middle or other non-intuitionistic statements, the more likely you are dealing with one of these. But it can be hard to have good intuition about this. In some cases it may be possible to ask "can I use this statement to prove for an arbitrary proposition" (see ordtriexmid for example ), but this is not always an easy technique to apply.
• Switching between 2th and 2false might help (e.g. dfnul2, dfnul3, rab0).
• In many cases statements which are equivalent in classical logic become several non-equivalent statements (e.g. exclusive or, ordinals, non-empty versus inhabited, apartness versus negated equality). This is usually a good place to look for a literature reference, but don't be afraid to change the statement being proved to "what you really meant is X" as appropriate.
• If a statement has multiple equivalences in set.mm (e.g. mo2 and mo3 , or dffun2 and dffun3 ) and only some of them in iset.mm, sometimes a pretty similar proof will work (that is, which one to use in the original proof may have been a fairly arbitrary choice).
• A number of theorems related to functions (especially ovex and fvex ) in set.mm perform case elimination based on whether we are evaluating the function within its domain or outside it. The most straightforward solution is to use fnovex or funfvex which only work within the domain. Using these may involve rearranging logic, for example by changing rexlimivw to rexlimdva (example: ovelrn or indeed most uses of fnovex and funfvex). If a function value is inhabited, we know we are evaluating it within its domain by relelfvdm.
• With excluded middle, and are equivalent (where is an ordinal). In such theorems, is generally the more interesting condition constructively.
• Reverse closure in set.mm uses excluded middle (for example ovrcl or ndmfvrcl ). The most general way to handle this is to add more conditions that we are evaluating operations within their domains (for example set.mm's addasspi versus iset.mm's addasspig in which conditions such as are added, or set.mm's ltbtwnnq versus iset.mm's ltbtwnnqq, in which is changed to ). If the result of applying a function is inhabited, then we know we applied it within its domain - that is relelfvdm or elmpt2cl may be useful.
• With excluded middle not equal () and apart (#) are equivalent. When working with real and complex numbers, apartness is almost always what you want. See df-ap for more on apartness.
• Exclusive or ( ) is equivalent to given excluded middle but we just have one direction (xorbin). Consider intuitionizing as (example: rpnegap).
• If you get stuck, ask! (for example in a GitHub issue or on the mailing list). We have a number of contributors who have experience in constructive mathematics in general, or iset.mm in particular, and one of the best things about doing/learning mathematics in metamath is the collaborative nature of how we develop it.

Metamath Proof Explorer cross reference

This is a list of theorems from the Metamath Proof Explorer (which assumes the law of the excluded middle throughout) which we do not have in the Intuitionistic Logic Explorer (generally because they are not provable without the law of the excluded middle, although some could be proved but aren't for a variety of reasons), together with the closest replacements.

set.mm iset.mm notes
ax-3 , con4d , con34b , necon4bd con3 The form of contraposition which removes negation does not hold in intutionistic logic.
pm2.18 pm2.01 See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.
pm2.18d , pm2.18i pm2.01d See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.
notnotrd , notnotri , notnotr , notnotb notnot Double negation introduction holds but not double negation elimination.
mt3d mtod
nsyl2 nsyl
pm2.61 , pm2.61d , pm2.61d1 , pm2.61d2 , pm2.61i , pm2.61ii , pm2.61nii , pm2.61iii , pm2.61ian , pm2.61dan , pm2.61ddan , pm2.61dda , pm2.61ine , pm2.61ne , pm2.61dne , pm2.61dane , pm2.61da2ne , pm2.61da3ne , pm2.61iine none
dfbi1 , dfbi3 df-bi, dfbi2
impcon4bid, con4bid, notbi, con1bii, con4bii, con2bii con3, condc
xor3 , nbbn xorbin, xornbi, xor3dc, nbbndc
biass biassdc
df-or , pm4.64 , pm2.54 , orri , orrd pm2.53, ori, ord
imor , imori imorr, imorri, imordc
pm4.63 pm3.2im
iman imanim
annim annimim
oran oranim
ianor pm3.14
biluk bilukdc
ecase3d none This is a form of case elimination.
dedlem0b dedlemb
pm4.42 pm4.42r
3ianor 3ianorr
df-nan and other theorems using NAND (Sheffer stroke) notation none A quick glance at the internet shows this mostly being used in the presence of excluded middle; in any case it is not currently present in iset.mm
df-xor df-xor Although the definition of exclusive or is called df-xor in both set.mm and iset.mm (at least currently), the definitions are not equivalent (in the absence of excluded middle).
xnor none The set.mm proof uses theorems not in iset.mm.
xorass none The set.mm proof uses theorems not in iset.mm.
xor2 xoranor, xor2dc
xornan xor2dc
xornan2 none See discussion under df-nan
xorneg2 , xorneg1 , xorneg none The set.mm proofs use theorems not in iset.mm.
xorexmid none A form of excluded middle
df-ex exalim
alex alexim
exnal exnalim
alexn alexnim
exanali exanaliim
19.35 , 19.35ri 19.35-1
19.30 none
19.39 i19.39
19.24 i19.24
19.36 , 19.36v 19.36-1
19.37 , 19.37v 19.37-1
19.32 19.32r
19.31 19.31r
exdistrf exdistrfor
exmo exmonim
mo2 mo2r, mo3
nne nner, nnedc
exmidne dcne
neor pm2.53, ori, ord
neorian pm3.14
nnel none The reverse direction could be proved; the forward direction is double negation elimination.
nfrald nfraldxy, nfraldya
rexnal rexnalim
rexanali none
nrexralim none
dfral2 ralexim
dfrex2 rexalim
nfrexd nfrexdxy, nfrexdya
nfral nfralxy, nfralya
nfra2 nfra1, nfralya
nfrex nfrexxy, nfrexya
r19.30 none
r19.35 r19.35-1
ralcom2 ralcom
nss nssr
sspss sspssr
n0f n0rf
n0 n0r, fin0, fin0or
neq0 neq0r
reximdva0 reximdva0m
rabn0 rabn0m, rabn0r
ssdif0 ssdif0im
inssdif0 inssdif0im
undif1 undif1ss
undif2 undif2ss
inundif inundifss
undif undifss
ssundif ssundifim
uneqdifeq uneqdifeqim
r19.2z r19.2m
r19.45zv r19.45mv
dfif2 df-if
ifsb none
dfif4 none Unused in set.mm
dfif5 none Unused in set.mm
ifnot none
ifan none
ifor none
ifeq1da, ifeq2da none
ifclda none
ifeqda none
elimif , ifbothda , ifid , eqif , ifval , elif , ifel , ifeqor , 2if2 , ifcomnan , csbif , csbifgOLD none
ifboth ifbothdc
ifcl , ifcld ifcldcd
dedth , dedth2h , dedth3h , dedth4h , dedth2v , dedth3v , dedth4v , elimhyp , elimhyp2v , elimhyp3v , elimhyp4v , elimel , elimdhyp , keephyp , keephyp2v , keephyp3v , keepel , ifex , ifexg none Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
difsnid difsnss One direction, for any set
nndifsnid for a natural number
fidifsnid for a finite set
iundif2 iundif2ss
iindif2 iindif2m
iinin2 iinin2m
iinin1 iinin1m
iinvdif iindif2m Unused in set.mm
riinn0 riinm
riinrab iinrabm
iinuni iinuniss
iununi iununir
rintn0 rintm
trintss trintssm
trint0 trint0m
ax-rep ax-coll There are a lot of ways to state replacement and most/all of them hold, for example zfrep6 or funimaexg.
csbexg , csbex csbexga, csbexa set.mm uses case elimination to remove the condition.
intex inteximm, intexr inteximm is the forward direction (but for inhabited rather than non-empty classes) and intexr is the reverse direction.
intexab intexabim
intexrab intexrabim
iinexg iinexgm Changes not empty to inhabited
intabs none Lightly used in set.mm, and the set.mm proof is not intuitionistic
moabex euabex In general, most of the set.mm theorems still hold, but a decent number of the ones get caught up on "there are two cases: the set exists or it does not"
snex snexg, snex The iset.mm version of snex has an additional hypothesis
nssss nssssr
opex opexg, opex The iset.mm version of opex has additional hypotheses
df-so df-iso Although we define to describe a weakly linear order (such as real numbers), there are some orders which are also trichotomous, for example nntri3or, pitri3or, and nqtri3or.
sotric sotricim One direction, for any weak linear order.
sotritric For a trichotomous order.
nntri2 For the specific order
pitric For the specific order
nqtric For the specific order
sotrieq sotritrieq For a trichotomous order
sotrieq2 see sotrieq and then apply ioran
issoi issod, ispod Many of the set.mm usages of issoi don't carry over, so there is less need for this convenience theorem.
isso2i issod Presumably this could be proved if we need it.
df-fr df-frind
fri , dffr2 , frc none That any subset of the base set has an element which is minimal accordng to a well-founded relation presumably implies excluded middle (or is otherwise unprovable).
frss freq2 Because the definition of is different than set.mm, the proof would need to be different.
frirr frirrg We do not yet have a lot of theorems for the case where is a proper class.
fr2nr none Shouldn't be hard to prove if we need it (using a proof similar to frirrg and en2lp).
frminex none Presumably unprovable.
efrn2lp none Should be easy but lightly used in set.mm
dfepfr , epfrc none Presumably unprovable.
df-we df-wetr
wess none See frss entry. Holds for (see for example wessep).
weso wepo
wecmpep none does not imply trichotomy in iset.mm
wefrc , wereu , wereu2 none Presumably not provable
dmxpid dmxpm
relimasn imasng
opswap opswapg
cnvso cnvsom
tz7.7 none
ordelssne none
ordelpss none
ordsseleq , onsseleq onelss, eqimss, nnsseleq Taken together, onelss and eqimss represent the reverse direction of the biconditional from ordsseleq . For natural numbers the biconditional is provable.
ordtri3or nntri3or Ordinal trichotomy implies the law of the excluded middle as shown in ordtriexmid.
ordtri2 nntri2 ordtri2 for all ordinals presumably implies excluded middle although we don't have a specific proof analogous to ordtriexmid.
ordtri3 , ordtri4 , ordtri2or3 , dford2 none Ordinal trichotomy implies the law of the excluded middle as shown in ordtriexmid. We don't have similar proofs for every variation of of trichotomy but most of them are presumably powerful enough to imply excluded middle.
ordtri1 , ontri1 , onssneli , onssnel2i ssnel, nntri1 ssnel is a trichotomy-like theorem which does hold, although it is an implication whereas ordtri1 is a biconditional. nntri1 is biconditional, but just for natural numbers.
ordtr2 , ontr2 none Implies excluded middle as shown at ontr2exmid
ordtr3 none This is weak linearity of ordinals, which presumably implies excluded middle by ordsoexmid.
ord0eln0 , on0eln0 ne0i, nn0eln0
nsuceq0 nsuceq0g
ordsssuc trsucss
ordequn none If you know which ordinal is larger, you can achieve a similar result via theorems such as oneluni or ssequn1.
ordun onun2
ordtri2or none Implies excluded middle as shown at ordtri2orexmid.
ordtri2or2 nntri2or2 ordtri2or2 implies excluded middle as shown at ordtri2or2exmid.
onsseli none See entry for ordsseleq
unizlim none The reverse direction is basically uni0 plus limuni
on0eqel 0elnn The full on0eqel is conjectured to imply excluded middle by an argument similar to ordtriexmid
snsn0non none Presumably would be provable (by first proving as in the set.mm proof, and then using that to show that is not a transitive set).
onxpdisj none Unused in set.mm
onnev none Presumably provable (see snsn0non entry)
iotaex iotacl, euiotaex
dffun3 dffun5r
dffun5 dffun5r
dffv3 dffv3g
fvex funfvex when evaluating a function within its domain
fvexg, fvex when the function is a set and is evaluated at a set
relrnfvex when evaluating a relation whose range is a set
mptfvex when the function is defined via maps-to, yields a set for all inputs, and is evaluated at a set
1stexg, 2ndexg for the functions and
ndmfv ndmfvg The case is fvprc.
elfvdm relelfvdm
elfvex relelfvdm
f0cli ffvelrn
dff3 dff3im
dff4 dff4im
fvunsn fvunsng
funiunfv fniunfv, funiunfvdm
funiunfvf funiunfvdmf
riotaex riotacl, riotaexg
csbriota , csbriotagOLD csbriotag
riotaxfrd none Although it may be intuitionizable, it is lightly used in set.mm.
ovex fnovex when the operation is a function evaluated within its domain.
ovexg when the operation is a set and is evaluated at a set
relrnfvex when the operation is a relation whose range is a set
mpt2fvex When the operation is defined via maps-to, yields a set on any inputs, and is being evaluated at two sets.
fnov fnovim
ov3 ovi3 Although set.mm's ov3 could be proved, it is only used a few places in set.mm, and in iset.mm those places need the modified form found in ovi3.
ndmovg , ndmov ndmfvg These theorems are generally used in set.mm for case elimination which is why we just have the general ndmfvg rather than an operation-specific version.
ndmovcl , ndmovcom , ndmovass , ndmovdistr , ndmovord , and ndmovordi none These theorems are generally used in set.mm for case elimination and the most straightforward way to avoid them is to add conditions that we are evaluating functions within their domains.
ndmovrcl elmpt2cl, relelfvdm
caov4 caov4d Note that caov4d has a closure hypothesis.
caov411 caov411d Note that caov411d has a closure hypothesis.
caov42 caov42d Note that caov42d has a closure hypothesis.
caovdir caovdird caovdird adds some constraints about where the operations are evaluated.
caovdilem caovdilemd
caovlem2 caovlem2d
caovmo caovimo
ofval fnofval
offn , offveq , caofid0l , caofid0r , caofid1 , caofid2 none Assuming we really need to add conditions that the operations are functions being evaluated within their domains, there would be a fair bit of intuitionizing.
ordeleqon none
ssonprc none not provable (we conjecture), but interesting enough to intuitionize anyway. is provable, and is provable. (Why isn't df-pss stated so that the set difference is inhabited? If so, you could prove .)
onint onintss onint implies excluded middle as shown in onintexmid.
onint0 none Thought to be "trivially not intuitionistic", and it is not clear if there is an alternate way to state it that is true. If the empty set is in A then of course |^| A = (/), but the converse seems difficult. I don't know so much about the structure of the ordinals without linearity,
onssmin, onminesb, onminsb none Conjectured to not be provable without excluded middle, for the same reason as onint.
oninton onintonm
onintrab none The set.mm proof relies on the converse of inteximm.
onintrab2 onintrab2im The converse would appear to need the converse of inteximm.
oneqmin none Falls as written because it implies onint, but it might be useful to keep the reverse direction for subsets that do have a minimum.
onminex none
onmindif2 none Conjectured to not be provable without excluded middle.
onmindif2 none Conjectured to not be provable without excluded middle.
ordpwsuc ordpwsucss See the ordpwsucss comment for discussion of the succcessor-like properites of . Full ordpwsuc implies excluded middle as seen at ordpwsucexmid.
ordsucelsuc onsucelsucr, nnsucelsuc The converse of onsucelsucr implies excluded middle, as shown at onsucelsucexmid.
ordsucsssuc onsucsssucr, nnsucsssuc The converse of onsucsssucr implies excluded middle, as shown at onsucsssucexmid.
ordsucuniel sucunielr Full ordsucuniel implies excluded middle, as shown at ordsucunielexmid.
ordsucun none yet Conjectured to be provable in the reverse direction, but not the forward direction (implies some order totality).
ordunpr none Presumably not provable without excluded middle.
ordunel none Conjectured to not be provable (ordunel implies ordsucun).
onsucuni, ordsucuni none Conjectured to not be provable without excluded middle.
orduniorsuc none Presumably not provable.
ordunisuc onunisuci, unisuc, unisucg
orduniss2 onuniss2
0elsuc none This theorem may appear to be innocuous but it implies excluded middle as shown at 0elsucexmid.
onuniorsuci none Conjectured to not be provable without excluded middle.
onuninsuci, orduninsuc none Conjectured to be provable in the forward direction but not the reverse one.
ordunisuc2 ordunisuc2r

The forward direction is conjectured to imply excluded middle. Here is a sketch of the proposed proof.

Let om' be the set of all finite iterations of suc' A = on . (We can formalize this proof but not until we have om and at least finite induction.) Then om' = U. om' because if x e. om' then x = suc'^n (/) for some n, and then x C_ suc'^n (/) implies x e. suc'^(n+1) (/) e. om' so x e. U. om'.

Now supposing the theorem, we know that A. x e. om' suc x e. om', so in particular 2o e. om', that is, 2o = suc'^n (/) for some n. (Note that 1o = suc' (/) - see pw0.) For n = 0 and n = 1 this is clearly false, and for n = m+3 we have 1o e. suc' suc' (/) , so 2o C_ suc' suc' (/), so 2o e. suc' suc' suc' (/) C_ suc' suc' suc' suc'^m (/) = 2o, contradicting ordirr.

Thus 2o = suc' suc' (/) = suc' 1o. Applying this to X = we have X C_ 1o implies X e. suc' 1o = 2o and hence X = (/) \/ X = 1o, and LEM follows (by ordtriexmidlem2 for and rabsnt as seen in the onsucsssucexmid proof for ).

ordzsl, onzsl, dflim3, nlimon none
dflim4 df-ilim We conjecture that dflim4 is not equivalent to df-ilim.
limsuc none yet Conjectured to be provable.
limsssuc none yet Conjectured to be provable.
tfinds tfis3
findsg uzind4 findsg presumably could be proved, but there hasn't been a need for it.
xpexr2 xpexr2m
1stval 1stvalg
2ndval 2ndvalg
1stnpr none May be intuitionizable, but very lightly used in set.mm.
2ndnpr none May be intuitionizable, but very lightly used in set.mm.
brtpos brtposg
ottpos ottposg
ovtpos ovtposg
pwuninel pwuninel2 The set.mm proof of pwuninel uses case elimination.
iunonOLD iunon
smofvon2 smofvon2dm
tfr1 tfri1
tfr2 tfri2
tfr3 tfri3
tfr2b , recsfnon , recsval none These transfinite recursion theorems are lightly used in set.mm.
df-rdg df-irdg This definition combines the successor and limit cases (rather than specifying them as separate cases in a way which relies on excluded middle). In the words of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic", "we can still define many of the familiar set-theoretic operations by transfinite recursion on ordinals (see Aczel and Rathjen 2001, Section 4.2). This is fine as long as the definitions by transfinite recursion do not make case distinctions such as in the classical ordinal cases of successor and limit."
rdgfnon rdgifnon
ordge1n0 ordge1n0im, ordgt0ge1
ondif1 dif1o In set.mm, ondif1 is used for Cantor Normal Form
ondif2 , dif20el none The set.mm proof is not intuitionistic
brwitnlem none The set.mm proof is not intuitionistic
om0r om0, nnm0r
om00 nnm00
om00el none
suc11reg suc11g
frfnom frecfnom frecfnom adopts the frec notation and adds conditions on the characteristic function and initial value.
fr0g frec0g frec0g adopts the frec notation and adds a condition on the characteristic function.
frsuc frecsuc frecsuc adopts the frec notation and adds conditions on the characteristic function and initial value.
om0x om0
oaord1 none yet
oaword oawordi The other direction presumably could be proven but isn't yet.
omwordi nnmword The set.mm proof of omwordi relies on case elimination.
omword1 nnmword
nnawordex nnaordex nnawordex is only used a few places in set.mm
swoso none Unused in set.mm.
ecdmn0 ecdmn0m
erdisj, qsdisj, qsdisj2, uniinqs none These could presumably be restated to be provable, but they are lightly used in set.mm
xpider xpiderm
iiner iinerm
riiner riinerm
brecop2 none This is a form of reverse closure and uses excluded middle in its proof.
erov , erov2 none Unused in set.mm.
eceqoveq none Unused in set.mm.
df-sdom , relsdom , brsdom , dfdom2 , sdomdom , sdomnen , brdom2 , bren2 , domdifsn none Many aspects of strict dominance as developed in set.mm rely on excluded middle and a different definition would be needed if we wanted strict dominance to have the expected properties.
en1b en1bg
mapsnen , map1 , pw2f1olem , pw2f1o , pw2eng , pw2en none We have not added set exponentiation to iset.mm yet.
snfi snfig
difsnen fidifsnen
undom none The set.mm proof uses undif2 and we just have undif2ss
xpdom3 xpdom3m
domunsncan none The set.mm proof relies on difsnen
omxpenlem , omxpen , omf1o none The set.mm proof relies on omwordi
enfixsn none The set.mm proof relies on difsnen
sbth and its lemmas, sbthb , sbthcl fisbth The Schroeder-Bernstein Theorem implies excluded middle
2pwuninel 2pwuninelg
php phpm
snnen2o snnen2og, snnen2oprc
onomeneq , onfin , onfin2 none The set.mm proofs rely on excluded middle
nnsdomo , sucdom2 , sucdom , 0sdom1dom , 1sdom2 , sdom1 none iset.mm doesn't yet have strict dominance
modom , modom2 none The set.mm proofs rely on excluded middle
1sdom , unxpdom , unxpdom2 , sucxpdom none iset.mm doesn't yet have strict dominance
pssinf none The set.mm proof relies on sdomnen
fisseneq none The set.mm proof relies on excluded middle
ominf none Although this theorem presumably could be proved, it would probably need a very different proof than the set.mm one
isinf none The set.mm proof uses the converse of ssdif0im
fineqv none The set.mm proof relies on theorems we don't have, and even for the theorems we do have, we'd need to carefully look at what axioms they rely on.
pssnn none The set.mm proof uses excluded middle.
ssnnfi none The proof in ssfiexmid would apply to this as well as to ssfi , since
ssfi none Implies excluded middle as shown at ssfiexmid
domfi none The set.mm proof is in terms of ssfi
xpfir none Nonempty would need to be changed to inhabited, but the set.mm proof also uses domfi
infi none Presumably the proof of ssfiexmid could be adapted to show this implies excluded middle
rabfi none Presumably the proof of ssfiexmid could be adapted to show this implies excluded middle
finresfin none The set.mm proof is in terms of ssfi
f1finf1o none The set.mm proof is not intuitionistic
nfielex none The set.mm proof relies on neq0
en1eqsn , en1eqsnbi none The set.mm proof relies on fisseneq
diffi diffisn, diffifi diffi is not provable, as shown at diffitest
enp1ilem , enp1i , en2 , en3 , en4 none The set.mm proof relies on excluded middle and undif1
findcard3 none The set.mm proof is in terms of strict dominance.
unfi none Not provable according to Remark 8.1.17 of [AczelRathjen].
ax-reg , axreg2 , zfregcl ax-setind ax-reg implies excluded middle as seen at regexmid
df-rank and all theorems realted to the rank function none One possible definition is Definition 9.3.4 of [AczelRathjen], p. 91
df-aleph and all theorems involving aleph none
df-cf and all theorems involving cofinality none
df-acn and all theorems using this definition none
cardf2 , cardon , isnum2 cardcl, isnumi It would also be easy to prove if there is a need.
ennum none The set.mm proof relies on isnum2
tskwe none Relies on df-sdom
xpnum none The set.mm proof relies on isnum2
cardval3 cardval3ex
cardid2 none The set.mm proof relies on onint
isnum3 none
oncardid none The set.mm proof relies on cardid2
cardidm none Presumably this would need a condition on but even with that, the set.mm proof relies on cardid2
oncard none The set.mm proof relies on theorems we don't have, and this theorem is unused in set.mm.
ficardom none Both the set.mm proof, and perhaps some possible alternative proofs, rely on onomeneq and perhaps other theorems we don't have currently.
ficardid none The set.mm proof relies on cardid2
cardnn none The set.mm proof relies on a variety of theorems we don't have currently.
cardnueq0 none The set.mm proof relies on cardid2
cardne none The set.mm proof relies on ordinal trichotomy (and if that can be solved there might be some more minor problems which require revisions to the theorem)
carden2a none The set.mm proof relies on excluded middle.
carden2b carden2bex
card1 , cardsn none Rely on a variety of theorems we don't currently have. Lightly used in set.mm.
carddomi2 none The set.mm proof relies on excluded middle.
sdomsdomcardi none Relies on a variety of theorems we don't currently have.
entri3 fientri3 Because full entri3 is equivalent to the axiom of choice, it implies excluded middle.
df-wina , df-ina , df-tsk , df-gru , ax-groth and all theorems related to inaccessibles and large cardinals none For an introduction to inaccessibles and large set properties see Chapter 18 of [AczelRathjen], p. 165 (including why "large set properties" is more apt terminology than "large cardinal properties" in the absence of excluded middle).
mulcompi mulcompig
mulasspi mulasspig
distrpi distrpig
mulcanpi mulcanpig
ltapi ltapig
ltmpi ltmpig
nlt1pi nlt1pig
df-nq df-nqqs
df-nq df-nqqs
df-erq none Not needed given df-nqqs
df-plq df-plqqs
df-mq df-mqqs
df-1nq df-1nqqs
df-ltnq df-ltnqqs
elpqn none Not needed given df-nqqs
ordpipq ordpipqqs
addnqf dmaddpq, addclnq It should be possible to prove that is a function, but so far there hasn't been a need to do so.
mulcomnq mulcomnqg
mulassnq mulassnqg
recmulnq recmulnqg
ltanq ltanqg
ltmnq ltmnqg
ltexnq ltexnqq
archnq archnqq
df-np df-inp
df-1p df-i1p
df-plp df-iplp
df-ltp df-iltp
elnp , elnpi elinp
prn0 prml, prmu
elprnq elprnql, elprnqu
prcdnq prcdnql, prcunqu
prub prubl
prnmax prnmaxl
npomex none
genpv genipv
genpcd genpcdl
genpnmax genprndl
ltrnq ltrnqg, ltrnqi
genpass genpassg
plpv plpvlu
mpv mpvlu
nqpr nqprlu
mulclprlem mulnqprl, mulnqpru
mulcompr mulcomprg
mulasspr mulassprg
distrlem1pr distrlem1prl, distrlem1pru
distrlem4pr distrlem4prl, distrlem4pru
distrlem5pr distrlem5prl, distrlem5pru
distrpr distrprg
ltprord ltprordil There hasn't yet been a need to investigate versions which are biconditional or which involve proper subsets.
psslinpr ltsopr
prlem934 prarloc2
ltexprlem1 , ltexprlem2 , ltexprlem3 , ltexprlem4 none See the lemmas for ltexpri generally.
ltexprlem5 ltexprlempr
ltexprlem6 ltexprlemfl, ltexprlemfu
ltexprlem7 ltexprlemrl, ltexprlemru
ltapr ltaprg
prlem936 prmuloc2
reclem2pr recexprlempr
reclem3pr recexprlem1ssl, recexprlem1ssu
reclem4pr recexprlemss1l, recexprlemss1u, recexprlemex
supexpr , suplem1pr , suplem2pr caucvgprpr The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
mulcmpblnrlem mulcmpblnrlemg
ltsrpr ltsrprg
dmaddsr , dmmulsr none Although these presumably could be proved in a way similar to dmaddpq and dmmulpq (in fact dmaddpqlem would seem to be easily generalizable to anything of the form ), we haven't yet had a need to do so.
mulcomsr mulcomsrg
mulasssr mulasssrg
distrsr distrsrg
ltasr ltasrg
sqgt0sr mulgt0sr, apsqgt0
recexsr recexsrlem This would follow from sqgt0sr (as in the set.mm proof of recexsr), but "not equal to zero" would need to be changed to "apart from zero".
mappsrpr , ltpsrpr , map2psrpr prsrpos, prsrlt, srpospr
supsrlem , supsr caucvgsr The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
axaddf , ax-addf , axmulf , ax-mulf none Because these are described as deprecated in set.mm, we haven't figured out what would be involved in proving them for iset.mm.
ax1ne0 , ax-1ne0 ax0lt1, ax-0lt1, 1ap0, 1ne0
axrrecex , ax-rrecex axprecex, ax-precex
axpre-lttri , ax-pre-lttri axpre-ltirr, axpre-ltwlin, ax-pre-ltirr, ax-pre-ltwlin
axpre-sup , ax-pre-sup , axsup axcaucvg, ax-caucvg, caucvgre The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. We express completeness using sequences.
elimne0 none Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
xrltnle xrlenlt
ssxr df-xr Lightly used in set.mm
ltnle , ltnlei , ltnled lenlt, zltnle
lttri2 , lttri4 ztri3or, qtri3or Real number trichotomy is not provable.
leloe , eqlelt , leloei , leloed , eqleltd none
leltne , leltned leltap, leltapd
ltneOLD ltne, ltap
letric , letrii , letrid zletric, qletric
ltlen , ltleni , ltlend ltleap, zltlen, qltlen
ne0gt0 , ne0gt0d ap0gt0, ap0gt0d
lecasei , ltlecasei none These are real number trichotomy
lelttric zlelttric, qlelttric
lttri2i none This can be read as "two real numbers are non-equal if and only if they are apart" which relies on excluded middle
lttrid , lttri2d , lttri4d none These are real number trichotomy
leneltd leltapd
dedekind , dedekindle none
mul02lem1 none The one use in set.mm is not needed in iset.mm.
negex negcl
msqgt0 , msqgt0i , msqgt0d apsqgt0 "Not equal to zero" is changed to "apart from zero"
relin01 none Relies on real number trichotomy.
ltordlem , ltord1 , leord1 , eqord1 , ltord2 , leord2 , eqord2 none These depend on real number trichotomy and are not used until later in set.mm.
wloglei , wlogle none These depend on real number trichotomy and are not used until later in set.mm.
recex recexap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcand, mulcan2d mulcanapd, mulcanap2d In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulcan , mulcan2 , mulcani mulcanap, mulcanap2, mulcanapi In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mul0or , mul0ori , mul0ord none Remark 2.19 of [Geuvers] says that this does not hold in general and has a counterexample.
mulne0 , mulne0i , mulne0d mulap0, mulap0i, mulap0d
receu receuap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
mulnzcnopr none
msq0i , msq0d sqeq0, sqeq0i These slight restatements of sqeq0 are unused in set.mm.
mulcan1g , mulcan2g various cancellation theorems Presumably this is unavailable for the same reason that mul0or is unavailable.
1div0 none This could be proved, but the set.mm proof does not work as-is.
divval divvalap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divmul , divmul2 , divmul3 divmulap, divmulap2, divmulap3 In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divcl , reccl divclap, recclap In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divcan1 , divcan2 divcanap1, divcanap2 In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
diveq0 diveqap0 In theorems involving reciprocals or division, not equal to zero changes to apart from zero.
divne0b , divne0 divap0b, divap0
recne0 recap0
recid , recid2 recidap, recidap2
divrec divrecap
divrec2 divrecap2
divass divassap
div23 , div32 , div13 , div12 div23ap, div32ap, div13ap, div12ap
divdir , divcan3 , divcan4 divdirap, divcanap3, divcanap4
div11 , divid , div0 div11ap, dividap, div0ap
diveq1 , divneg , divsubdir diveqap1, divnegap, divsubdirap
recrec , rec11 , rec11r recrecap, rec11ap, rec11rap
divmuldiv , divdivdiv , divcan5 divmuldivap, divdivdivap, divcanap5
divmul13 , divmul24 , divmuleq divmul13ap, divmul24ap, divmuleqap
recdiv , divcan6 , divdiv32 , divcan7 recdivap, divcanap6, divdiv32ap, divcanap7
dmdcan , divdiv1 , divdiv2 , recdiv2 dmdcanap, divdivap1, divdivap2, recdivap2
conjmul , rereccl, redivcl conjmulap, rerecclap, redivclap
div2neg , divneg2 div2negap, divneg2ap
recclzi , recne0zi , recidzi recclapzi, recap0apzi, recidapzi
reccli , recidi , recreci recclapi, recidapi, recrecapi
dividi , div0i dividapi, div0api
divclzi , divcan1zi , divcan2zi divclapzi, divcanap1zi, divcanap2zi
divreczi , divcan3zi , divcan4zi divrecapzi, divcanap3zi, divcanap4zi
rec11i , rec11ii rec11api, rec11apii
divcli , divcan2i , divcan1i , divreci , divcan3i , divcan4i divclapi, divcanap2i, divcanap1i, divrecapi, divcanap3i, divcanap4i
div0i divap0i
divasszi , divmulzi , divdirzi , divdiv23zi divassapzi, divmulapzi, divdirapzi, divdiv23apzi
divmuli , divdiv32i divmulapi, divdiv32api
divassi , divdiri , div23i , div11i divassapi, divdirapi, div23api, div11api
rerecclzi , rereccli , redivclzi , redivcli rerecclapzi, rerecclapi, redivclapzi, redivclapi
reccld , rec0d , recidd , recid2d , recrecd , dividd , div0d recclapd, recap0d, recidapd, recidap2d, recrecapd, dividapd, div0apd
divcld , divcan1d , divcan2d , divrecd , divrec2d , divcan3d , divcan4d divclapd, divcanap1d, divcanap2d, divrecapd, divrecap2d, divcanap3d, divcanap4d
diveq0d , diveq1d , diveq1ad , diveq0ad , divne1d , div0bd , divnegd , divneg2d , div2negd diveqap0d, diveqap1d, diveqap1ad, diveqap0ad, divap1d, divap0bd, divnegapd, divneg2apd, div2negapd
divne0d , recdivd , recdiv2d , divcan6d , ddcand , rec11d divap0d, recdivapd, recdivap2d, divcanap6d, ddcanapd, rec11apd
divmuld , div32d , div13d , divdiv32d , divcan5d , divcan5rd , divcan7d , dmdcand , dmdcan2d , divdiv1d , divdiv2d divmulapd, div32apd, div13apd, divdiv32apd, divcanap5d, divcanap5rd, divcanap7d, dmdcanapd, dmdcanap2d, divdivap1d, divdivap2d
divmul2d, divmul3d, divassd, div12d, div23d, divdird, divsubdird, div11d divmulap2d, divmulap3d, divassapd, div12apd, div23apd, divdirapd, divsubdirapd, div11apd
divmuldivd divmuldivapd
rereccld , redivcld rerecclapd, redivclapd
mvllmuld mvllmulapd
elimgt0 , elimge0 none Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.
mulge0b , mulle0b , mulsuble0b none Presumably unprovable for reasons analogous to mul0or.
ledivp1i , ltdivp1i none Presumably could be proved, but unused in set.mm.
crne0 crap0
ofsubeq0 , ofnegsub , ofsubge0 none Depend on ofval and/or offn .
df-nn dfnn2
dfnn3 dfnn2 Presumably could be proved, as it is a slight variation of dfnn2
avgle none
nnunb none Presumably provable from arch but unused in set.mm.
frnnn0supp , frnnn0fsupp nn0supp iset.mm does not yet have either the notation, or in some cases the theorems, related to the support of a function or a fintely supported function.
suprzcl none
zriotaneg none Lightly used in set.mm
suprfinzcl none
decex deccl
uzwo , uzwo2 , nnwo , nnwof , nnwos none Presumably would imply excluded middle, unless there is something which makes this different from nnregexmid.
uzinfmi , nninfm , nn0infm , infmssuzle , infmssuzcl none We do not yet have a supremum notation, or most supremum theorems, in iset.mm
negn0 negm
supminf none We do not yet have a supremum notation, or most supremum theorems, in iset.mm
zsupss , suprzcl2 , suprzub , uzsupss none We do not yet have a supremum notation, or most supremum theorems, in iset.mm
uzwo3 , zmin none Proved in terms of supremum theorems and presumably not possible without excluded middle.
zbtwnre none Proved in terms of supremum theorems and presumably not possible without excluded middle.
rebtwnz qbtwnz
rpneg rpnegap
xrlttri , xrlttri2 none A generalization of real trichotomy.
xrleloe , xrleltne , dfle2 none Consequences of real trichotomy.
xrltlen none We presumably could prove an analogue to ltleap but we have not yet defined apartness for extended reals (# is for complex numbers).
dflt2 none
xrletri none
xrmax1 , xrmax2 , xrmin1 , xrmin2 , xrmaxeq , xrmineq , xrmaxlt , xrltmin , xrmaxle , xrlemin , max1 , max2 , min1 , min2 , maxle , lemin , maxlt , ltmin , max0sub , ifle none
qbtwnxr none Presumably provable from qbtwnre.
qsqueeze none yet Presumably provable from qbtwnre and squeeze0, but unused in set.mm.
qextltlem , qextlt , qextle none The set.mm proof is not intuitionistic.
xralrple , alrple none yet If we had qbtwnxr , it looks like the set.mm proof would work with minor changes.
xnegex xnegcl
ixxub , ixxlb none
iccen none
supicc , supiccub , supicclub , supicclub2 none
ixxun , ixxin none
ioo0 none Once we have qbtwnxr , it may be possible to rearrange the logic from set.mm so that this proof works (via rabeq0, ralnex, and xrlenlt perhaps).
ioon0 none Presumably non-empty would need to be changed to inhabited, see also discussion at ioo0
iooid iooidg
ndmioo none See discussion at ndmov but set.mm uses excluded middle, both in proving this and in using it.
lbioo , ubioo lbioog, ubioog
iooin none
ico0 , ioc0 none Possibly similar to ioo0 ; see discussion there.
icc0 icc0r
ioorebas ioorebasg
ge0xaddcl , ge0xmulcl none Rely on xaddcl and xmulcl ; see discussion in this list for those theorems.
icoun , snunioo , snunico , snunioc , prunioo none
ioojoin none
difreicc none
iccsplit none This depends, apparently in an essential way, on real number trichotomy.
xov1plusxeqvd none This presumably could be proved if not equal is changed to apart, but is lightly used in set.mm.
fzn0 fzm
fz0 none Although it would be possible to prove a version of this with the additional conditions that and , the theorem is lightly used in set.mm.
fzon0 fzom
fzo0n0 fzo0m
ssfzoulel none Presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
fzonfzoufzol none Presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
elfznelfzo , elfznelfzob , injresinjlem , injresinj none Some or all of this presumably could be proven, but the set.mm proof is not intuitionistic and it is lightly used in set.mm.
flcl , reflcl , flcld flqcl, flqcld
fllelt flqlelt
flle flqle
flltp1 , fllep1 flqltp1
fraclt1 , fracle1 qfraclt1
fracge0 qfracge0
flge flqge
fllt flqlt
flflp1 none The set.mm proof relies on case elimination.
flidm flqidm
flidz flqidz
flltnz flqltnz
flwordi flqwordi
flword2 flqword2
flval2 , flval3 none Unused in set.mm
flbi flqbi
flbi2 flqbi2
ico01fl0 none Presumably could be proved for rationals, but lightly used in set.mm.
flge0nn0 flqge0nn0
flge1nn flqge1nn
refldivcl flqcl
flmulnn0 flqmulnn0
fldivle flqle
ltdifltdiv none Unused in set.mm.
fldiv4lem1div2uz2 , fldiv4lem1div2 none Presumably provable, but lightly used in set.mm.
ceilval ceilqval The set.mm ceilval, with a real argument and no additional conditions, is probably provable if there is a need.
dfceil2 , ceilval2 none Unused in set.mm.
ceicl ceiqcl
ceilcl ceilqcl
ceilge ceilqge
ceige ceiqge
ceim1l ceiqm1l
ceilm1lt ceilqm1lt
ceile ceiqle
ceille ceilqle
ceilidz ceilqidz
flleceil flqleceil
fleqceilz flqeqceilz
quoremz , quoremnn0 , quoremnn0ALT none Unused in set.mm.
intfrac2 intqfrac2
fldiv flqdiv
fldiv2 none Presumably would be provable if real is changed to rational.
fznnfl none Presumably would be provable if real is changed to rational.
uzsup , ioopnfsup , icopnfsup , rpsup , resup , xrsup none As with most theorems involving supremums, these would likely need significant changes
modval modqval As with theorems such as flqcl, we prove most of the modulo related theorems for rationals, although other conditions on real arguments other than whether they are rational would be possible in the future.
modvalr modqvalr
modcl , modcld modqcl, modqcld
flpmodeq flqpmodeq
mod0 modq0
mulmod0 mulqmod0
negmod0 negqmod0
modge0 modqge0
modlt modqlt
modelico modqelico
moddiffl modqdiffl
moddifz modqdifz
modfrac modqfrac
flmod flqmod
intfrac intqfrac
modmulnn modqmulnn
om2uz0i frec2uz0d
om2uzsuci frec2uzsucd
om2uzuzi frec2uzuzd
om2uzlti frec2uzltd
om2uzlt2i frec2uzlt2d
om2uzrani frec2uzrand
om2uzf1oi frec2uzf1od
om2uzisoi frec2uzisod
om2uzoi , ltweuz , ltwenn , ltwefz none We do not have a syntax for well ordering and based on theorems like nnregexmid, there is probably little room for this concept.
om2uzrdg frec2uzrdg
uzrdglem frecuzrdglem
uzrdgfni frecuzrdgfn
uzrdg0i frecuzrdg0
uzrdgsuci frecuzrdgsuc
uzinf none See ominf
uzrdgxfr none Presumably could be proved if restated in terms of frec (a la frec2uz0d). However, it is lightly used in set.mm.
fzennn frecfzennn
fzen2 frecfzen2
cardfz none Cardinality does not work the same way without excluded middle and iset.mm has few cardinality related theorems.
hashgf1o frechashgf1o
fzfi fzfig
fzfid fzfigd
fzofi fzofig
fsequb none Seems like it might be provable, but unused in set.mm
fsequb2 none The set.mm proof does not work as-is
fseqsupcl , fseqsupubi none iset.mm does not have the "sup" syntax and lacks most supremum theorems from set.mm.
uzindi none This could presumably be proved, perhaps from uzind4, but is lightly used in set.mm
axdc4uz none Although some versions of constructive mathematics accept dependent choice, we have not yet developed it in iset.mm
ssnn0fi , rabssnn0fi none Conjectured to imply excluded middle along the lines of nnregexmid or ssfiexmid
df-seq df-iseq
seqex iseqex The only difference is the differing syntax of .
seqeq1 , seqeq2 , seqeq3 , seqeq1d , seqeq2d , seqeq3d , seqeq123d iseqeq1, iseqeq2, iseqeq3
nfseq nfiseq
seqval iseqval
seqfn iseqfn
seq1 , seq1i iseq1
seqp1 , seqp1i iseqp1
seqm1 iseqm1
seqcl2 none yet Presumably could prove this (with adjustments analogous to iseqp1). The third argument to in iset.mm would correspond to rather than (some of the other related theorems do not allow and to be different, but seqcl2 does).
seqf2 none yet Presumably could prove this, analogously to seqcl2.
seqcl iseqcl iseqcl requires that be defined on not merely as in seqcl . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqf iseqf
seqfveq2 iseqfveq2
seqfeq2 iseqfeq2
seqfveq iseqfveq
seqfeq iseqfeq
seqshft2 iseqshft2
seqres none Should be intuitionizable as with the other theorems, but unused in set.mm
serf iserf
serfre iserfre
sermono isermono isermono requires that be defined on not merely as in sermono . Given that the intention is to use this for infinite series, there may be no need to look into whether this requirement can be relaxed.
seqsplit iseqsplit iseqsplit requires that be defined on not merely as in seqsplit . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seq1p iseq1p iseq1p requires that be defined on not merely as in seq1p . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr3 iseqcaopr3 iseqcaopr3 requires that , , and be defined on not merely as in seqcaopr3 . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr2 iseqcaopr2 iseqcaopr2 requires that , , and be defined on not merely as in seqcaopr2 . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqcaopr iseqcaopr iseqcaopr requires that , , and be defined on not merely as in seqcaopr . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqf1o none May be adaptable with sufficient attention to issues such as whether the operation is closed with respect to as well as , and perhaps other issues.
seradd iseradd iseradd requires that , , and be defined on not merely as in seradd . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
sersub isersub isersub requires that , , and be defined on not merely as in sersub . This is not a problem when used on infinite sequences, but perhaps this requirement could be relaxed if there is a need.
seqid3 iseqid3, iseqid3s iseqid3 and iseqid3s differ with respect to which entries in need to be zero. Further refinements to the hypotheses including where needs to be defined and the like might be possible (with a greater amount of work), but perhaps these are sufficient.
seqid iseqid
seqid2 none yet It should be possible to come up with a (presumably modified) versiona of this, but we have not done so yet.
seqhomo iseqhomo
seqz , seqfeq4 , seqfeq3 none yet It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
seqdistr iseqdistr
ser0 iser0 The only difference is the syntax of .
ser0f iser0f The only difference is the syntax of .
serge0 serige0 This theorem uses as the final argument to which probably will be more convenient even if all the values are elements of a subset of .
serle serile This theorem uses as the final argument to which probably will be more convenient even if all the values are elements of a subset of .
ser1const , seqof , seqof2 none yet It should be possible to come up with some (presumably modified) versions of these, but we have not done so yet.
df-exp df-iexp The difference is that in iset.mm takes three arguments compared with two in set.mm.
expval expival
expnnval expinnval
expneg expnegap0 The set.mm theorem does not exclude the case of dividing by zero.
expneg2 expineg2
expn1 expn1ap0
expcl2lem expcl2lemap
reexpclz reexpclzap
expclzlem expclzaplem
expclz expclzap
expne0 expap0
expne0i expap0i
expnegz expnegzap
mulexpz mulexpzap
exprec exprecap
expmulz expmulzap
expsub expsubap
expp1z expp1zap
expm1 expm1ap
expdiv expdivap
expcan , expcand none Presumably provable, but the set.mm proof uses eqord1 and the theorem is lightly used in set.mm
ltexp2 , leexp2 , leexp2 , ltexp2d , leexp2d none Presumably provable, but the set.mm proof uses ltord1
ltexp2r , ltexp2rd none Presumably provable, but the set.mm proof uses ltexp2
sqdiv sqdivap
sqgt0 sqgt0ap
iexpcyc none yet See discussion under df-mod ; modulus for integers would suffice so issues with modulus for reals would not be an impediment.
sqrecii , sqrecd exprecap
sqdivi sqdivapi
sqgt0i sqgt0api
sqlecan lemul1 Unused in set.mm
sqeqori none The reverse direction is oveq1 together with sqneg. The forward direction is presumably not provable, see mul0or for more discussion.
subsq0i , sqeqor none Variations of sqeqori .
sq01 none Lightly used in set.mm. Presumably not provable as stated, for reasons analogous to mul0or .
crreczi none Presumably could be proved if not-equal is changed to apart, but unused in set.mm.
expmulnbnd none Should be possible to prove this or something similar, but the set.mm proof relies on case elimination based on whether or not.
digit2 , digit1 none Depends on modulus and floor, and unused in set.mm.
modexp none Depends on modulus. Presumably it, or something similar, can be made to work as it is mostly about integers rather than reals.
discr1 , discr none The set.mm proof uses real number trichotomy.
sqrecd sqrecapd
expclzd expclzapd
exp0d expap0d
expnegd expnegapd
exprecd exprecapd
expp1zd expp1zapd
expm1d expm1apd
expsubd expsubapd
sqdivd sqdivapd
expdivd expdivapd
reexpclzd reexpclzapd
sqgt0d sqgt0apd
seqshft none currently need to figure out how to adjust this for the differences between set.mm and iset.mm concerning .
df-sgn and theorems related to the sgn function none To choose a value near zero requires knowing the argument with unlimited precision. It would be possible to define for rational numbers, or real numbers apart from zero.
mulre mulreap
rediv redivap
imdiv imdivap
cjdiv cjdivap
sqeqd none The set.mm proof is not intuitionistic, and this theorem is unused in set.mm.
cjdivi cjdivapi
cjdivd cjdivapd
redivd redivapd
imdivd imdivapd
df-sqrt df-rsqrt See discussion of complex square roots in the comment of df-rsqrt. Here's one possibility if we do want to define square roots on (some) complex numbers: It should be possible to define the complex square root function on all complex numbers satisfying # , using a similar construction to the one used in set.mm. You need the real square root as a basis for the construction, but then there is a trick using the complex number x + |x| (see sqreu) that yields the complex square root whenever it is apart from zero (you need to divide by it at one point IIRC), which is exactly on the negative real line. You can either live with this constraint, which gives you the complex square root except on the negative real line (which puts a hole at zero), or you can extend it by continuity to zero as well by joining it with the real square root. The disjunctive domain of the resulting function might not be so useful though.
sqrtval sqrtrval See discussion of complex square roots in the comment of df-rsqrt
01sqrex and its lemmas resqrex Both set.mm and iset.mm prove resqrex although via different mechanisms so there is no need for 01sqrex.
cnpart none See discussion of complex square roots in the comment of df-rsqrt
sqrmo rsqrmo See discussion of complex square roots in the comment of df-rsqrt
resqreu rersqreu Although the set.mm theorem is primarily about real square roots, the iset.mm equivalent removes some complex number related parts.
sqrtneg , sqrtnegd none Although it may be possible to extend the domain of square root somewhat beyond nonnegative reals without excluded middle, in general complex square roots are difficult, as discussed in the comment of df-rsqrt
sqrtm1 none Although it may be possible to extend the domain of square root somewhat beyond nonnegative reals without excluded middle, in general complex square roots are difficult, as discussed in the comment of df-rsqrt
absrpcl , absrpcld absrpclap, absrpclapd
absdiv , absdivzi , absdivd absdivap, absdivapzi, absdivapd
absor , absori , absord none We could prove this for rationals, or real numbers apart from zero, if we wanted
absmod0 none See df-mod ; we may want to supply this for rationals or integers
absexpz absexpzap
max0add none This would hold if we defined the maximum of two real numbers and recast this theoerem in terms of that (rather than using in a way which relies on real number trichotomy).
absz none Although this is presumably provable, the set.mm proof is not intuitionistic and it is lightly used in set.mm
recval recvalap
absgt0 , absgt0i absgt0ap, absgt0api
absmax none The right hand side of this theorem (which is in terms of absolute value) could serve as a definition of maximum, but the left hand side (in terms of ) is not well behaved in the absence of trichotomy.
abs1m none Because this theorem provides as the answer if and as the answer if , and uses excluded middle to combine those cases, it is presumably not provable as stated. We could prove the theorem with the additional condition that # , but it is unused in set.mm.
abslem2 none Although this could presumably be proved if not equal were changed to apart, it is lightly used in set.mm.
rddif , absrdbnd none As described under df-fl , floor is problematic without excluded middle.
rexanre none yet May be feasible once we've added maximum for real numbers
rexfiuz none Relies on findcard2
rexuzre none Presumably could be intuitionized although it would be easier after we add maximum for real numbers. It is unused in set.mm for whatever that is worth.
rexico none May be feasible once we've added maximum for real numbers
caubnd none The caubnd proof uses theorems related to finite sets and maximums which are not present in iset.mm, so this will need developing those areas and/or a different approach than set.mm.
sqreulem , sqreu , sqrtcl , sqrtcld , sqrtthlem , sqrtf , sqrtth , sqsqrtd , msqsqrtd , sqr00d , sqrtrege0 , sqrtrege0d , eqsqrtor , eqsqrtd , eqsqrt2d rersqreu, resqrtcl, resqrtcld, resqrtth As described at df-rsqrt, square roots of complex numbers are in set.mm defined with the help of excluded middle.
df-limsup and all superior limit theorems none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
df-rlim and theorems related to limits of partial functions on the reals none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to convergence.
df-o1 and theorems related to eventually bounded functions none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
df-lo1 and theorems related to eventually upper bounded functions none This is not developed in iset.mm currently. If it was it would presumably be noticeably different from set.mm given various differences relating to sequence convergence, supremums, etc.
serclim0 iserclim0 The only difference is the syntax of .
reccn2 none yet Will need to be revamped to deal with negated equality versus apartness and perhaps other issues.
clim2ser clim2iser The only difference is the syntax of .
clim2ser2 clim2iser2 The only difference is the syntax of .
iserex iiserex The only difference is the syntax of .
isermulc2 iisermulc2 The only difference is the syntax of .
iserle iserile The only difference is the syntax of .
iserge0 iserige0 The only difference is the syntax of .
climserle climserile The only difference is the syntax of .
isershft none yet Relies on seqshft
isercoll and its lemmas, isercoll2 none yet The set.mm proof would need modification
climsup none To even show convergence would presumably require a hypothesis related to the rate of convergence. Not to mention the lack of much in the way of supremum notation or theorems in iset.mm.
climbdd none Presumably could be proved but the current proof of caubnd would need at least some minor adjustments.
caurcvg2 climrecvg1n
caucvg climcvg1n
caucvgb climcaucn, climcvg1n Without excluded middle, there are additional complications related to the rate of convergence.
serf0 serif0 The only difference is the syntax of .
iseralt none The set.mm proof relies on caurcvg2 which does not specify a rate of convergence.
sumex none This will need to replaced by suitable closure theorems.
sumeq2w sumeq2 Presumably could be proved, and perhaps also would rely only on extensionality (and logical axioms). But unused in set.mm.
seq1st none The second argument to , at least as handled in theorems such as iseqfn, must be defined on all integers greater than or equal to , not just at itself. It may be possible to patch this up, but seq1st is unused in set.mm.
algr0 ialgr0 Several hypotheses are tweaked or added to reflect differences in how works
algrf ialgrf Several hypotheses are tweaked or added to reflect differences in how works
algrp1 ialgrp1 Several hypotheses are tweaked or added to reflect differences in how works
alginv ialginv Two hypotheses are tweaked or added to reflect differences in how works
algcvg ialgcvg as defined in df-iseq has a third argument
algcvga ialgcvga Two hypotheses are tweaked or added to reflect differences in how works
algfx ialgfx Two hypotheses are tweaked or added to reflect differences in how works

Bibliography
1. [AczelRathjen] Aczel, Peter, and Rathjen, Michael, "Constructive Set Theory", 2018, http://www1.maths.leeds.ac.uk/~rathjen/book.pdf
2. [Apostol] Apostol, Tom M., Calculus, vol. 1, 2nd ed., John Wiley & Sons Inc. (1967) [QA300.A645 1967].
3. [ApostolNT] Apostol, Tom M., Introduction to analytic number theory, Springer-Verlag, New York, Heidelberg, Berlin (1976) [QA241.A6].
4. [Bauer] Bauer, Andrej, "Five stages of accepting constructive mathematics," Bulletin (New Series) of the American Mathematical Society, 54:481-498 (2017), DOI: 10.1090/bull/1556 .
5. [BauerTaylor] Andrej Bauer and Paul Taylor, "The Dedekind Reals in Abstract Stone Duality", Mathematical structures in computer science, 19(4):757–838, 2009, http://www.paultaylor.eu/ASD/dedras.pdf
6. [BellMachover] Bell, J. L., and M. Machover, A Course in Mathematical Logic, North-Holland, Amsterdam (1977) [QA9.B3953].
7. [Bobzien] Bobzien, Susanne, "Stoic Logic", The Cambridge Companion to Stoic Philosophy, Brad Inwood (ed.), Cambridge University Press (2003-2006), https://philpapers.org/rec/BOBSL.
8. [BourbakiEns] Bourbaki, Nicolas, Théorie des ensembles, Springer-Verlag, Berlin Heidelberg (2007); available in English (for purchase) at http://www.springer.com/us/book/9783540225256 (retrieved 15-Aug-2016). Page references in iset.mm are for the French edition.
9. [BourbakiTop1] Bourbaki, Nicolas, Topologie générale, Chapitres 1 à 4, Springer-Verlag, Berlin Heidelberg (2007); available in English (for purchase) at http://www.springer.com/us/book/9783540642411 (retrieved 15-Aug-2016). Page references in set.mm are for the French edition.
10. [ChoquetDD] Choquet-Bruhat, Yvonne and Cecile DeWitt-Morette, with Margaret Dillard-Bleick, Analysis, Manifolds and Physics, Elsevier Science B.V., Amsterdam (1982) [QC20.7.A5C48 1981].
11. [Crosilla] Crosilla, Laura, "Set Theory: Constructive and Intuitionistic ZF", The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/sum2015/entries/set-theory-constructive/.
12. [Moschovakis] Moschovakis, Joan, "Intuitionistic Logic", The Stanford Encyclopedia of Philosophy (Spring 2015 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2015/entries/logic-intuitionistic/.
13. [Eisenberg] Eisenberg, Murray, Axiomatic Theory of Sets and Classes, Holt, Rinehart and Winston, Inc., New York (1971) [QA248.E36].
14. [Enderton] Enderton, Herbert B., Elements of Set Theory, Academic Press, Inc., San Diego, California (1977) [QA248.E5].
15. [Gleason] Gleason, Andrew M., Fundamentals of Abstract Analysis, Jones and Bartlett Publishers, Boston (1991) [QA300.G554].
16. [Geuvers] Herman Geuvers, Randy Pollack, Freek Wiedijk, Jan Zwanenburg, "Skeleton for the Proof development leading to the Fundamental Theorem of Algebra", October 2, 2000, http://www.cs.ru.nl/~herman/PUBS/FTA.mathproof.ps.gz (accessed 19 Feb 2020).
17. [Hamilton] Hamilton, A. G., Logic for Mathematicians, Cambridge University Press, Cambridge, revised edition (1988) [QA9.H298 1988].
18. [Heyting] Heyting, A., Intuitionism: An introduction, North-Holland publishing company, Amsterdam, second edition (1966).
19. [Hitchcock] Hitchcock, David, The peculiarities of Stoic propositional logic, McMaster University; available at http://www.humanities.mcmaster.ca/~hitchckd/peculiarities.pdf (retrieved 3 Jul 2016).
20. [HoTT] The Univalent Foundations Program, Homotopy Type Theory: Univalent Foundations of Mathematics, https://homotopytypetheory.org/book, first edition.
21. [Jech] Jech, Thomas, Set Theory, Academic Press, San Diego (1978) [QA248.J42].
22. [KalishMontague] Kalish, D. and R. Montague, "On Tarski's formalization of predicate logic with identity," Archiv für Mathematische Logik und Grundlagenforschung, 7:81-101 (1965) [QA.A673].
23. [Kreyszig] Kreysig, Erwin, Introductory Functional Analysis with Applications, John Wiley & Sons, New York (1989) [QA320.K74].
24. [Kunen] Kunen, Kenneth, Set Theory: An Introduction to Independence Proofs, Elsevier Science B.V., Amsterdam (1980) [QA248.K75].
25. [KuratowskiMostowski] Kuratowski, K. and A. Mostowski, Set Theory: with an Introduction to Descriptive Set Theory, 2nd ed., North-Holland, Amsterdam (1976) [QA248.K7683 1976].
26. [Levy] Levy, Azriel, Basic Set Theory, Dover Publications, Mineola, N.Y. (2002) [QA248.L398 2002].
27. [Lopez-Astorga] Lopez-Astorga, Miguel, "The First Rule of Stoic Logic and its Relationship with the Indemonstrables", Revista de Filosofía Tópicos (2016); available at http://www.scielo.org.mx/pdf/trf/n50/n50a1.pdf (retrieved 3 Jul 2016).
28. [Margaris] Margaris, Angelo, First Order Mathematical Logic, Blaisdell Publishing Company, Waltham, Massachusetts (1967) [QA9.M327].
29. [Megill] Megill, N., "A Finitely Axiomatized Formalization of Predicate Calculus with Equality," Notre Dame Journal of Formal Logic, 36:435-453 (1995) [QA.N914]; available at http://projecteuclid.org/euclid.ndjfl/1040149359 (accessed 11 Nov 2014); the PDF preprint has the same content (with corrections) but pages are numbered 1-22, and the database references use the numbers printed on the page itself, not the PDF page numbers.
30. [Mendelson] Mendelson, Elliott, Introduction to Mathematical Logic, 2nd ed., D. Van Nostrand (1979) [QA9.M537].
31. [Monk1] Monk, J. Donald, Introduction to Set Theory, McGraw-Hill, Inc. (1969) [QA248.M745].
32. [Monk2] Monk, J. Donald, "Substitutionless Predicate Logic with Identity," Archiv für Mathematische Logik und Grundlagenforschung, 7:103-121 (1965) [QA.A673].
33. [Quine] Quine, Willard van Orman, Set Theory and Its Logic, Harvard University Press, Cambridge, Massachusetts, revised edition (1969) [QA248.Q7 1969].
34. [Sanford] Sanford, David H., If P, then Q: Conditionals and the Foundations of Reasoning, 2nd ed., Routledge Taylor & Francis Group (2003); ISBN 0-415-28369-8; available at https://books.google.com/books?id=h_AUynB6PA8C&pg=PA39#v=onepage&q&f=false (retrieved 3 Jul 2016).
35. [Schechter] Schechter, Eric, Handbook of Analysis and Its Foundations, Academic Press, San Diego (1997) [QA300.S339].
36. [Stoll] Stoll, Robert R., Set Theory and Logic, Dover Publications, Inc. (1979) [QA248.S7985 1979].
37. [Suppes] Suppes, Patrick, Axiomatic Set Theory, Dover Publications, Inc. (1972) [QA248.S959].
38. [TakeutiZaring] Takeuti, Gaisi, and Wilson M. Zaring, Introduction to Axiomatic Set Theory, Springer-Verlag, New York, second edition (1982) [QA248.T136 1982].
39. [Tarski] Tarski, Alfred, "A Simplified Formalization of Predicate Logic with Identity," Archiv für Mathematische Logik und Grundlagenforschung, 7:61-79 (1965) [QA.A673].
40. [WhiteheadRussell] Whitehead, Alfred North, and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, Cambridge, 1962 [QA9.W592 1962].