P. 94 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdcnulALT 9301	Alternate proof of bdcnul 9300. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 9279, or use the corresponding characterizations of its elements followed by bdelir 9282. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED (/)
Theorem	bdeq0 9302	Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED x = (/)
Theorem	bj-bd0el 9303	Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED (/) e. x
Theorem	bdcpw 9304	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>   |- BOUNDED ~PA
Theorem	bdcsn 9305	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED {x }
Theorem	bdcpr 9306	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED {x , y }
Theorem	bdctp 9307	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED {x , y , z }
Theorem	bdsnss 9308*	Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>   |- BOUNDED {x } C_ A
Theorem	bdvsn 9309*	Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x = {y }
Theorem	bdop 9310	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <.x , y >.
Theorem	bdcuni 9311	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U.x
Theorem	bdcint 9312	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^|x
Theorem	bdciun 9313*	The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>   |- BOUNDED U_x e. y A
Theorem	bdciin 9314*	The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>   |- BOUNDED |^|_x e. y A
Theorem	bdcsuc 9315	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 9316*	Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED x = suc y
Theorem	bj-bdsucel 9317	Boundedness of the formula "the successor of the setvar x belongs to the setvar y". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED suc x e. y
Theorem	bdcriota 9318*	A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
|- BOUNDED ph   &   |- E!x e. y ph   =>   |- BOUNDED ( iota_x e. y ph)
5.3.6  Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 9319*	Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3866. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>   |- A. aE. bA.x(x e. b <-> (x e. a /\ ph))
Theorem	bdsep2 9320*	Version of ax-bdsep 9319 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>   |- E. bA.x(x e. b <-> (x e. a /\ ph))
Theorem	bdsepnft 9321*	Closed form of bdsepnf 9322. Version of ax-bdsep 9319 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 19-Oct-2019.)
|- BOUNDED ph   =>   |- (A.x F/ bph -> E. bA.x(x e. b <-> (x e. a /\ ph)))
Theorem	bdsepnf 9322*	Version of ax-bdsep 9319 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 9323. (Contributed by BJ, 5-Oct-2019.)
|- F/ bph   &   |- BOUNDED ph   =>   |- E. bA.x(x e. b <-> (x e. a /\ ph))
Theorem	bdsepnfALT 9323*	Alternate proof of bdsepnf 9322, not using bdsepnft 9321. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ bph   &   |- BOUNDED ph   =>   |- E. bA.x(x e. b <-> (x e. a /\ ph))
Theorem	bdzfauscl 9324*	Closed form of the version of zfauscl 3868 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
|- BOUNDED ph   =>   |- (A e. V -> E.yA.x(x e. y <-> (x e. A /\ ph)))
Theorem	bdbm1.3ii 9325*	Bounded version of bm1.3ii 3869. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &   |- E.xA.y(ph -> y e. x)   =>   |- E.xA.y(y e. x <-> ph)
Theorem	bj-nalset 9326*	nalset 3878 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E.xA.y y e. x
Theorem	bj-vprc 9327	vprc 3879 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 9328	nvel 3880 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 9329	vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E.x x = _V
Theorem	bdinex1 9330	Bounded version of inex1 3882. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &   |- A e. _V   =>   |- (A i^i B) e. _V
Theorem	bdinex2 9331	Bounded version of inex2 3883. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &   |- A e. _V   =>   |- (B i^i A) e. _V
Theorem	bdinex1g 9332	Bounded version of inex1g 3884. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>   |- (A e. V -> (A i^i B) e. _V)
Theorem	bdssex 9333	Bounded version of ssex 3885. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &   |- B e. _V   =>   |- (A C_ B -> A e. _V)
Theorem	bdssexi 9334	Bounded version of ssexi 3886. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &   |- B e. _V   &   |- A C_ B   =>   |- A e. _V
Theorem	bdssexg 9335	Bounded version of ssexg 3887. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>   |- ((A C_ B /\ B e. C) -> A e. _V)
Theorem	bdssexd 9336	Bounded version of ssexd 3888. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- (ph -> B e. C)   &   |- (ph -> A C_ B)   &   |- BOUNDED A   =>   |- (ph -> A e. _V)
Theorem	bdrabexg 9337*	Bounded version of rabexg 3891. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &   |- BOUNDED A   =>   |- (A e. V -> {x e. A | ph } e. _V)
Theorem	bj-inex 9338	The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ((A e. V /\ B e. W) -> (A i^i B) e. _V)
Theorem	bj-intexr 9339	vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^|A e. _V -> A =/= (/))
Theorem	bj-intnexr 9340	vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^|A = _V -> -. |^|A e. _V)
Theorem	bj-zfpair2 9341	Proof of zfpair2 3936 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- {x , y } e. _V
Theorem	bj-prexg 9342	Proof of prexg 3938 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ((A e. V /\ B e. W) -> {A , B } e. _V)
Theorem	bj-snexg 9343	snexg 3927 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- (A e. V -> {A } e. _V)
Theorem	bj-snex 9344	snex 3928 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- A e. _V   =>   |- {A } e. _V
Theorem	bj-sels 9345*	If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
|- (A e. V -> E.x A e. x)
Theorem	bj-axun2 9346*	axun2 4138 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E.yA.z(z e. y <-> E.w(z e. w /\ w e. x))
Theorem	bj-uniex2 9347*	uniex2 4139 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E.y y = U.x
Theorem	bj-uniex 9348	uniex 4140 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>   |- U.A e. _V
Theorem	bj-uniexg 9349	uniexg 4141 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- (A e. V -> U.A e. _V)
Theorem	bj-unex 9350	unex 4142 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   &   |- B e. _V   =>   |- (A u. B) e. _V
Theorem	bdunexb 9351	Bounded version of unexb 4143. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &   |- BOUNDED B   =>   |- ((A e. _V /\ B e. _V) <-> (A u. B) e. _V)
Theorem	bj-unexg 9352	unexg 4144 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ((A e. V /\ B e. W) -> (A u. B) e. _V)
Theorem	bj-sucexg 9353	sucexg 4190 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- (A e. V -> suc A e. _V)
Theorem	bj-sucex 9354	sucex 4191 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>   |- suc A e. _V
5.3.6.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 9355	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
|- BOUNDED ph   =>   |- DECID ph
Theorem	bj-notbi 9356	Equivalence property for negation. TODO: minimize all theorems using notbid 591 and notbii 593. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
|- ((ph <-> ps) -> (-. ph <-> -. ps))
Theorem	bj-notbii 9357	Inference associated with bj-notbi 9356. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
|- (ph <-> ps)   =>   |- (-. ph <-> -. ps)
Theorem	bj-notbid 9358	Deduction form of bj-notbi 9356. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
|- (ph -> (ps <-> ch))   =>   |- (ph -> (-. ps <-> -. ch))
Theorem	bj-dcbi 9359	Equivalence property for DECID. TODO: solve conflict with dcbi 843; minimize dcbii 746 and dcbid 747 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
|- ((ph <-> ps) -> (DECID ph <-> DECID ps))
Theorem	bj-d0clsepcl 9360	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
|- DECID ph
5.3.6.2  Inductive classes and the class of natural numbers (finite ordinals)
Syntax	wind 9361	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 9362*	Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- (Ind A <-> ( (/) e. A /\ A.x e. A suc x e. A))
Theorem	bj-indsuc 9363	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
|- (Ind A -> (B e. A -> suc B e. A))
Theorem	bj-indeq 9364	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- (A = B -> (Ind A <-> Ind B))
Theorem	bj-bdind 9365	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 9366*	The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
|- Ind |^| {x e. A | Ind x }
Theorem	bj-dfom 9367	Alternate definition of om, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
|- om = |^| {x | Ind x }
Theorem	bj-omind 9368	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 9369	om is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- (A e. V -> (Ind A -> om C_ A))
Theorem	bj-ssom 9370*	A characterization of subclasses of om. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- (A.x(Ind x -> A C_ x) <-> A C_ om)
Theorem	bj-om 9371*	A set is equal to om if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- (A e. V -> (A = om <-> (Ind A /\ A.x(Ind x -> A C_ x))))
Theorem	bj-2inf 9372*	Two formulations of the axiom of infinity (see ax-infvn 9375 and bj-omex 9376) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E.x(Ind x /\ A.y(Ind y -> x C_ y)))
5.3.6.3  The first three Peano postulates The first three Peano postulates do not require the axiom of infinity. We give constructive proofs (only the proof of the second postulate has to be modified). We also prove a preliminary version of the fifth Peano postulate.
Theorem	bj-peano2 9373	Constructive proof of peano2 4261. Temporary note: another possibility is to simply replace sucexg 4190 with bj-sucexg 9353 in the proof of peano2 4261. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- (A e. om -> suc A e. om)
Theorem	peano5set 9374*	Version of peano5 4264 when om i^i 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.)
|- (( om i^i A) e. V -> (( (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A))
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 9375) and deduce that the class om of finite ordinals is a set (bj-omex 9376).
Axiom	ax-infvn 9375*	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 (omex 4259) using full separation that the wanted set exists. "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.)
|- E.x(Ind x /\ A.y(Ind y -> x C_ y))
Theorem	bj-omex 9376	Proof of omex 4259 from ax-infvn 9375. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
5.3.7.2  The remaining two Peano postulates In this section, we give constructive proofs of the remaining two (the fourth and fifth) Peano postulates. More precisely, we prove from the core axioms of CZF that the set of finite ordinals satisfies the Peano postulates and thus provides a model for the set of natural numbers.
Theorem	bdpeano5 9377*	Bounded version of peano5 4264. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>   |- (( (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)
Theorem	speano5 9378*	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 e. V /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)
5.3.7.3  Bounded induction In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction).
Theorem	findset 9379*	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 9380 for a proof when A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- (A e. V -> ((A C_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> A = om))
Theorem	bdfind 9380*	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 9379 for a proof when A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>   |- ((A C_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> A = om)
Theorem	bj-bdfindis 9381*	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 ph   &   |- F/xps   &   |- F/xch   &   |- F/xth   &   |- (x = (/) -> (ps -> ph))   &   |- (x = y -> (ph -> ch))   &   |- (x = suc y -> (th -> ph))   =>   |- ((ps /\ A.y e. om (ch -> th)) -> A.x e. om ph)
Theorem	bj-bdfindisg 9382*	Version of bj-bdfindis 9381 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9381 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &   |- F/xps   &   |- F/xch   &   |- F/xth   &   |- (x = (/) -> (ps -> ph))   &   |- (x = y -> (ph -> ch))   &   |- (x = suc y -> (th -> ph))   &   |- F/_xA   &   |- F/xta   &   |- (x = A -> (ph -> ta))   =>   |- ((ps /\ A.y e. om (ch -> th)) -> (A e. om -> ta))
Theorem	bj-bdfindes 9383	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 9381 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 ph   =>   |- (( [. (/) / x ].ph /\ A.x e. om (ph -> [. suc x / x ].ph)) -> A.x e. om ph)
Theorem	bj-nn0suc0 9384*	Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9394. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- (A e. om -> (A = (/) \/ E.x e. A A = suc x))
Theorem	bj-nntrans 9385	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- (A e. om -> (B e. A -> B C_ A))
Theorem	bj-nntrans2 9386	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- (A e. om -> Tr A)
Theorem	bj-nnelirr 9387	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 e. om -> -. A e. A)
Theorem	bj-nnen2lp 9388	A version of en2lp 4232 for natural numbers, which does not require ax-setind 4220. Note: using this theorem and bj-nnelirr 9387, 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 e. om /\ B e. om) -> -. (A e. B /\ B e. A))
Theorem	bj-peano4 9389	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 e. om /\ B e. om) -> ( suc A = suc B <-> A = B))
Theorem	bj-omtrans 9390	The set om is transitive. A natural number is included in om. The idea is to use bounded induction with the formula x C_ om. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with x C_ a and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- (A e. om -> A C_ om)
Theorem	bj-omtrans2 9391	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-omord 9392	The set om is an ordinal. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 9393	The set om is an ordinal. Constructive proof of omelon 4274. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-nn0suc 9394*	Proof of (biconditional form of) nn0suc 4270 from the core axioms of CZF. See also bj-nn0sucALT 9408. As a characterization of the elements of om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- (A e. om <-> (A = (/) \/ E.x e. om 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.
Theorem	setindft 9395*	Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
|- (A.x F/yph -> (A.x(A.y e. x [y / x]ph -> ph) -> A.xph))
Theorem	setindf 9396*	Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
|- F/yph   =>   |- (A.x(A.y e. x [y / x]ph -> ph) -> A.xph)
Theorem	setindis 9397*	Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
|- F/xps   &   |- F/xch   &   |- F/yph   &   |- F/yps   &   |- (x = z -> (ph -> ps))   &   |- (x = y -> (ch -> ph))   =>   |- (A.y(A.z e. y ps -> ch) -> A.xph)
Axiom	ax-bdsetind 9398*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
|- BOUNDED ph   =>   |- (A. a(A.y e. a [y / a]ph -> ph) -> A. aph)
Theorem	bdsetindis 9399*	Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &   |- F/xps   &   |- F/xch   &   |- F/yph   &   |- F/yps   &   |- (x = z -> (ph -> ps))   &   |- (x = y -> (ch -> ph))   =>   |- (A.y(A.z e. y ps -> ch) -> A.xph)
Theorem	bj-inf2vnlem1 9400*	Lemma for bj-inf2vn 9404. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- (A.x(x e. A <-> (x = (/) \/ E.y e. A x = suc y)) -> Ind A)