HomeHome Intuitionistic Logic Explorer
Theorem List (p. 95 of 95)
< Previous  Wrap >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 9401-9457   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
5.3.7  Axiom of infinity

In the absence of full separation, the axiom of infinity has to be stated more precisely, as the existence of the smallest class containing the empty set and the successor of each of its elements.

 
5.3.7.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 9401) and deduce that the class 𝜔 of finite ordinals is a set (bj-omex 9402).

 
Axiomax-infvn 9401* Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4254) from which one then proves, using full separation, that the wanted set exists (omex 4259). "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.)
x(Ind x y(Ind yxy))
 
Theorembj-omex 9402 Proof of omex 4259 from ax-infvn 9401. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
𝜔 V
 
5.3.7.2  Peano's fifth postulate

In this section, we give constructive proofs of two versions of Peano's fifth postulate.

 
Theorembdpeano5 9403* Bounded version of peano5 4264. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED A       ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
 
Theoremspeano5 9404* Version of peano5 4264 when A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
((A 𝑉 A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
 
5.3.7.3  Bounded induction and Peano's fourth postulate

In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction). We also prove Peano's fourth postulate. Together with the results from the previous sections, this proves from the core axioms of CZF (with infinity) that the set of finite ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.

 
Theoremfindset 9405* Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See bdfind 9406 for a proof when A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(A 𝑉 → ((A ⊆ 𝜔 A x A suc x A) → A = 𝜔))
 
Theorembdfind 9406* Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See findset 9405 for a proof when A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
BOUNDED A       ((A ⊆ 𝜔 A x A suc x A) → A = 𝜔)
 
Theorembj-bdfindis 9407* Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4266 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED φ    &   xψ    &   xχ    &   xθ    &   (x = ∅ → (ψφ))    &   (x = y → (φχ))    &   (x = suc y → (θφ))       ((ψ y 𝜔 (χθ)) → x 𝜔 φ)
 
Theorembj-bdfindisg 9408* Version of bj-bdfindis 9407 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9407 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED φ    &   xψ    &   xχ    &   xθ    &   (x = ∅ → (ψφ))    &   (x = y → (φχ))    &   (x = suc y → (θφ))    &   xA    &   xτ    &   (x = A → (φτ))       ((ψ y 𝜔 (χθ)) → (A 𝜔 → τ))
 
Theorembj-bdfindes 9409 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 9407 for explanations. From this version, it is easy to prove the bounded version of findes 4269. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED φ       (([∅ / x]φ x 𝜔 (φ[suc x / x]φ)) → x 𝜔 φ)
 
Theorembj-nn0suc0 9410* Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9424. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(A 𝜔 → (A = ∅ x A A = suc x))
 
Theorembj-nntrans 9411 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(A 𝜔 → (B ABA))
 
Theorembj-nntrans2 9412 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(A 𝜔 → Tr A)
 
Theorembj-nnelirr 9413 A natural number does not belong to itself. Version of elirr 4224 for natural numbers, which does not require ax-setind 4220. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(A 𝜔 → ¬ A A)
 
Theorembj-nnen2lp 9414 A version of en2lp 4232 for natural numbers, which does not require ax-setind 4220.

Note: using this theorem and bj-nnelirr 9413, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

((A 𝜔 B 𝜔) → ¬ (A B B A))
 
Theorembj-peano4 9415 Remove from peano4 4263 dependency on ax-setind 4220. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
((A 𝜔 B 𝜔) → (suc A = suc BA = B))
 
Theorembj-omtrans 9416 The set 𝜔 is transitive. A natural number is included in 𝜔. Constructive proof of elnn 4271.

The idea is to use bounded induction with the formula x ⊆ 𝜔. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with x𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

(A 𝜔 → A ⊆ 𝜔)
 
Theorembj-omtrans2 9417 The set 𝜔 is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr 𝜔
 
Theorembj-nnord 9418 A natural number is an ordinal. Constructive proof of nnord 4277. Can also be proved from bj-nnelon 9419 if the latter is proved from bj-omssonALT 9423. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(A 𝜔 → Ord A)
 
Theorembj-nnelon 9419 A natural number is an ordinal. Constructive proof of nnon 4275. Can also be proved from bj-omssonALT 9423. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(A 𝜔 → A On)
 
Theorembj-omord 9420 The set 𝜔 is an ordinal. Constructive proof of ordom 4272. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Ord 𝜔
 
Theorembj-omelon 9421 The set 𝜔 is an ordinal. Constructive proof of omelon 4274. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
𝜔 On
 
Theorembj-omsson 9422 Constructive proof of omsson 4278. See also bj-omssonALT 9423. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
𝜔 ⊆ On
 
Theorembj-omssonALT 9423 Alternate proof of bj-omsson 9422. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜔 ⊆ On
 
Theorembj-nn0suc 9424* Proof of (biconditional form of) nn0suc 4270 from the core axioms of CZF. See also bj-nn0sucALT 9438. As a characterization of the elements of 𝜔, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
 
5.3.8  Set induction

In this section, we add the axiom of set induction to the core axioms of CZF.

 
5.3.8.1  Set induction

In this section, we prove some variants of the axiom of set induction.

 
Theoremsetindft 9425* Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
(xyφ → (x(y x [y / x]φφ) → xφ))
 
Theoremsetindf 9426* Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
yφ       (x(y x [y / x]φφ) → xφ)
 
Theoremsetindis 9427* Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
xψ    &   xχ    &   yφ    &   yψ    &   (x = z → (φψ))    &   (x = y → (χφ))       (y(z y ψχ) → xφ)
 
Axiomax-bdsetind 9428* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
BOUNDED φ       (𝑎(y 𝑎 [y / 𝑎]φφ) → 𝑎φ)
 
Theorembdsetindis 9429* Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
BOUNDED φ    &   xψ    &   xχ    &   yφ    &   yψ    &   (x = z → (φψ))    &   (x = y → (χφ))       (y(z y ψχ) → xφ)
 
Theorembj-inf2vnlem1 9430* Lemma for bj-inf2vn 9434. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(x(x A ↔ (x = ∅ y A x = suc y)) → Ind A)
 
Theorembj-inf2vnlem2 9431* Lemma for bj-inf2vnlem3 9432 and bj-inf2vnlem4 9433. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(x A (x = ∅ y A x = suc y) → (Ind 𝑍u(𝑡 u (𝑡 A𝑡 𝑍) → (u Au 𝑍))))
 
Theorembj-inf2vnlem3 9432* Lemma for bj-inf2vn 9434. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED A    &   BOUNDED 𝑍       (x A (x = ∅ y A x = suc y) → (Ind 𝑍A𝑍))
 
Theorembj-inf2vnlem4 9433* Lemma for bj-inf2vn2 9435. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(x A (x = ∅ y A x = suc y) → (Ind 𝑍A𝑍))
 
Theorembj-inf2vn 9434* A sufficient condition for 𝜔 to be a set. See bj-inf2vn2 9435 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED A       (A 𝑉 → (x(x A ↔ (x = ∅ y A x = suc y)) → A = 𝜔))
 
Theorembj-inf2vn2 9435* A sufficient condition for 𝜔 to be a set; unbounded version of bj-inf2vn 9434. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(A 𝑉 → (x(x A ↔ (x = ∅ y A x = suc y)) → A = 𝜔))
 
Axiomax-inf2 9436* Another axiom of infinity in a constructive setting (see ax-infvn 9401). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
𝑎x(x 𝑎 ↔ (x = ∅ y 𝑎 x = suc y))
 
Theorembj-omex2 9437 Using bounded set induction and the strong axiom of infinity, 𝜔 is a set, that is, we recover ax-infvn 9401 (see bj-2inf 9397 for the equivalence of the latter with bj-omex 9402). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜔 V
 
Theorembj-nn0sucALT 9438* Alternate proof of bj-nn0suc 9424, also constructive but from ax-inf2 9436, hence requiring ax-bdsetind 9428. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(A 𝜔 ↔ (A = ∅ x 𝜔 A = suc x))
 
5.3.8.2  Full induction

In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.

 
Theorembj-findis 9439* Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 9407 for a bounded version not requiring ax-setind 4220. See finds 4266 for a proof in IZF. From this version, it is easy to prove of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
xψ    &   xχ    &   xθ    &   (x = ∅ → (ψφ))    &   (x = y → (φχ))    &   (x = suc y → (θφ))       ((ψ y 𝜔 (χθ)) → x 𝜔 φ)
 
Theorembj-findisg 9440* Version of bj-findis 9439 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 9439 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
xψ    &   xχ    &   xθ    &   (x = ∅ → (ψφ))    &   (x = y → (φχ))    &   (x = suc y → (θφ))    &   xA    &   xτ    &   (x = A → (φτ))       ((ψ y 𝜔 (χθ)) → (A 𝜔 → τ))
 
Theorembj-findes 9441 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 9439 for explanations. From this version, it is easy to prove findes 4269. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
(([∅ / x]φ x 𝜔 (φ[suc x / x]φ)) → x 𝜔 φ)
 
5.3.9  Strong collection

In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.

 
Axiomax-strcoll 9442* Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.)
𝑎(x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
 
Theoremstrcoll2 9443* Version of ax-strcoll 9442 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
(x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
 
Theoremstrcollnft 9444* Closed form of strcollnf 9445. Version of ax-strcoll 9442 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
(xy𝑏φ → (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ)))
 
Theoremstrcollnf 9445* Version of ax-strcoll 9442 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
𝑏φ       (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
 
TheoremstrcollnfALT 9446* Alternate proof of strcollnf 9445, not using strcollnft 9444. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏φ       (x 𝑎 yφ𝑏y(y 𝑏x 𝑎 φ))
 
5.3.10  Subset collection

In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.

 
Axiomax-sscoll 9447* Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.)
𝑎𝑏𝑐z(x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))
 
Theoremsscoll2 9448* Version of ax-sscoll 9447 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐z(x 𝑎 y 𝑏 φ𝑑 𝑐 y(y 𝑑x 𝑎 φ))
 
5.4  Mathbox for David A. Wheeler
 
5.4.1  Allsome quantifier

These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some").

In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like xφψ do not imply that φ is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem.

The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines.

I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it.

The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀!x(φψ), and when restricted (applied to a class) we allow ∀!x Aφ. The first symbol after the setvar variable must always be if it is the form applied to a class, and since cannot begin a wff, it is unambiguous. The looks like it would be a problem because φ or ψ might include implications, but any implication arrow within any wff must be surrounded by parentheses, so only the implication arrow of ∀! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax.

For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome.

 
Syntaxwalsi 9449 Extend wff definition to include "all some" applied to a top-level implication, which means ψ is true whenever φ is true, and there is at least least one x where φ is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff ∀!x(φψ)
 
Syntaxwalsc 9450 Extend wff definition to include "all some" applied to a class, which means φ is true for all x in A, and there is at least one x in A. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff ∀!x Aφ
 
Definitiondf-alsi 9451 Define "all some" applied to a top-level implication, which means ψ is true whenever φ is true and there is at least one x where φ is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
(∀!x(φψ) ↔ (x(φψ) xφ))
 
Definitiondf-alsc 9452 Define "all some" applied to a class, which means φ is true for all x in A and there is at least one x in A. (Contributed by David A. Wheeler, 20-Oct-2018.)
(∀!x Aφ ↔ (x A φ x x A))
 
Theoremalsconv 9453 There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
(∀!x(x Aφ) ↔ ∀!x Aφ)
 
Theoremalsi1d 9454 Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(φ → ∀!x(ψχ))       (φx(ψχ))
 
Theoremalsi2d 9455 Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(φ → ∀!x(ψχ))       (φxψ)
 
Theoremalsc1d 9456 Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(φ → ∀!x Aψ)       (φx A ψ)
 
Theoremalsc2d 9457 Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
(φ → ∀!x Aψ)       (φx x A)
    < Previous  Wrap >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9457
  Copyright terms: Public domain < Previous  Wrap >