P. 156 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15501-15600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdsepnf 15501*	Version of ax-bdsep 15497 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 15502. Use bdsep1 15498 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- F/ b ph   &    |- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnfALT 15502*	Alternate proof of bdsepnf 15501, not using bdsepnft 15500. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ b ph   &    |- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdzfauscl 15503*	Closed form of the version of zfauscl 4153 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
|- BOUNDED ph   =>    |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) )
Theorem	bdbm1.3ii 15504*	Bounded version of bm1.3ii 4154. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- E. x A. y ( ph -> y e. x )   =>    |- E. x A. y ( y e. x <-> ph )
Theorem	bj-axemptylem 15505*	Lemma for bj-axempty 15506 and bj-axempty2 15507. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4159 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 15506*	Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4158. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4159 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 15507*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 15506. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4159 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 15508*	nalset 4163 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x
Theorem	bj-vprc 15509	vprc 4165 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 15510	nvel 4166 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 15511	vnex 4164 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 15512	Bounded version of inex1 4167. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 15513	Bounded version of inex2 4168. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 15514	Bounded version of inex1g 4169. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 15515	Bounded version of ssex 4170. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 15516	Bounded version of ssexi 4171. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	bdssexg 15517	Bounded version of ssexg 4172. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	bdssexd 15518	Bounded version of ssexd 4173. (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 15519*	Bounded version of rabexg 4176. (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 15520	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 15521	intexr 4183 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	bj-intnexr 15522	intnexr 4184 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A = _V -> -. |^| A e. _V )
Theorem	bj-zfpair2 15523	Proof of zfpair2 4243 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- { x , y } e. _V
Theorem	bj-prexg 15524	Proof of prexg 4244 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 15525	snexg 4217 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ( A e. V -> { A } e. _V )
Theorem	bj-snex 15526	snex 4218 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- { A } e. _V
Theorem	bj-sels 15527*	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 15528*	axun2 4470 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )
Theorem	bj-uniex2 15529*	uniex2 4471 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y y = U. x
Theorem	bj-uniex 15530	uniex 4472 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- U. A e. _V
Theorem	bj-uniexg 15531	uniexg 4474 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> U. A e. _V )
Theorem	bj-unex 15532	unex 4476 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 15533	Bounded version of unexb 4477. (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 15534	unexg 4478 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 15535	sucexg 4534 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> suc A e. _V )
Theorem	bj-sucex 15536	sucex 4535 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- suc A e. _V
13.2.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 15537	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
|- BOUNDED ph   =>    |- DECID ph
Theorem	bj-d0clsepcl 15538	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
|- DECID ph
13.2.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 15539	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 15540*	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 15541	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 15542	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- ( A = B -> (Ind A <-> Ind B ) )
Theorem	bj-bdind 15543	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 15544*	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-indind 15545*	If A is inductive and B is "inductive in A", then ( A i^i B ) is inductive. (Contributed by BJ, 25-Oct-2020.)
|- ( (Ind A /\ ( (/) e. B /\ A. x e. A ( x e. B -> suc x e. B ) ) ) -> Ind ( A i^i B ) )
Theorem	bj-dfom 15546	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 15547	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 15548	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 15549*	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 15550*	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 15551*	Two formulations of the axiom of infinity (see ax-infvn 15554 and bj-omex 15555) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E. x (Ind x /\ A. y (Ind y -> x C_ y ) ) )
13.2.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4630 and peano3 4632 already show this. In this section, we prove bj-peano2 15552 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 15552	Constructive proof of peano2 4631. Temporary note: another possibility is to simply replace sucexg 4534 with bj-sucexg 15535 in the proof of peano2 4631. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> suc A e. om )
Theorem	peano5set 15553*	Version of peano5 4634 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 ) )
13.2.10  CZF: 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.
13.2.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 15554) and deduce that the class om of natural number ordinals is a set (bj-omex 15555).
Axiom	ax-infvn 15554*	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 4624) from which one then proves, using full separation, that the wanted set exists (omex 4629). "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 15555	Proof of omex 4629 from ax-infvn 15554. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
13.2.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 15556*	Bounded version of peano5 4634. (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 15557*	Version of peano5 4634 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 )
13.2.10.3  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 natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.
Theorem	findset 15558*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4635 for a nonconstructive proof of the general case. See bdfind 15559 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 15559*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4635 for a nonconstructive proof of the general case. See findset 15558 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 15560*	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 4636 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4636, finds2 4637, finds1 4638. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( 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 15561*	Version of bj-bdfindis 15560 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 15560 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   &    |- F/_ x A   &    |- F/ x ta   &    |- ( x = A -> ( ph -> ta ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> ( A e. om -> ta ) )
Theorem	bj-bdfindes 15562	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 15560 for explanations. From this version, it is easy to prove the bounded version of findes 4639. (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 15563*	Constructive proof of a variant of nn0suc 4640. For a constructive proof of nn0suc 4640, see bj-nn0suc 15577. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 15564	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 15565	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 15566	A natural number does not belong to itself. Version of elirr 4577 for natural numbers, which does not require ax-setind 4573. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 15567	A version of en2lp 4590 for natural numbers, which does not require ax-setind 4573. Note: using this theorem and bj-nnelirr 15566, one can remove dependency on ax-setind 4573 from nntri2 6552 and nndcel 6558; 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 15568	Remove from peano4 4633 dependency on ax-setind 4573. 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 15569	The set om is transitive. A natural number is included in om. Constructive proof of elnn 4642. 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 15570	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 15571	A natural number is an ordinal class. Constructive proof of nnord 4648. Can also be proved from bj-nnelon 15572 if the latter is proved from bj-omssonALT 15576. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 15572	A natural number is an ordinal. Constructive proof of nnon 4646. Can also be proved from bj-omssonALT 15576. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 15573	The set om is an ordinal class. Constructive proof of ordom 4643. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 15574	The set om is an ordinal. Constructive proof of omelon 4645. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 15575	Constructive proof of omsson 4649. See also bj-omssonALT 15576. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On
Theorem	bj-omssonALT 15576	Alternate proof of bj-omsson 15575. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om C_ On
Theorem	bj-nn0suc 15577*	Proof of (biconditional form of) nn0suc 4640 from the core axioms of CZF. See also bj-nn0sucALT 15591. 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 ) )
13.2.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
13.2.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 15578*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
|- ( A. x F/ y ph -> ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph ) )
Theorem	setindf 15579*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
|- F/ y ph   =>    |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )
Theorem	setindis 15580*	Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ y ph   &    |- F/ y ps   &    |- ( x = z -> ( ph -> ps ) )   &    |- ( x = y -> ( ch -> ph ) )   =>    |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )
Axiom	ax-bdsetind 15581*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
|- BOUNDED ph   =>    |- ( A. a ( A. y e. a [ y / a ] ph -> ph ) -> A. a ph )
Theorem	bdsetindis 15582*	Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED ph   &    |- F/ x ps   &    |- F/ x ch   &    |- F/ y ph   &    |- F/ y ps   &    |- ( x = z -> ( ph -> ps ) )   &    |- ( x = y -> ( ch -> ph ) )   =>    |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )
Theorem	bj-inf2vnlem1 15583*	Lemma for bj-inf2vn 15587. 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 )
Theorem	bj-inf2vnlem2 15584*	Lemma for bj-inf2vnlem3 15585 and bj-inf2vnlem4 15586. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) ) )
Theorem	bj-inf2vnlem3 15585*	Lemma for bj-inf2vn 15587. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- BOUNDED Z   =>    |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )
Theorem	bj-inf2vnlem4 15586*	Lemma for bj-inf2vn2 15588. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )
Theorem	bj-inf2vn 15587*	A sufficient condition for om to be a set. See bj-inf2vn2 15588 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )
Theorem	bj-inf2vn2 15588*	A sufficient condition for om to be a set; unbounded version of bj-inf2vn 15587. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
|- ( A e. V -> ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A = om ) )
Axiom	ax-inf2 15589*	Another axiom of infinity in a constructive setting (see ax-infvn 15554). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
|- E. a A. x ( x e. a <-> ( x = (/) \/ E. y e. a x = suc y ) )
Theorem	bj-omex2 15590	Using bounded set induction and the strong axiom of infinity, om is a set, that is, we recover ax-infvn 15554 (see bj-2inf 15551 for the equivalence of the latter with bj-omex 15555). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om e. _V
Theorem	bj-nn0sucALT 15591*	Alternate proof of bj-nn0suc 15577, also constructive but from ax-inf2 15589, hence requiring ax-bdsetind 15581. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. om <-> ( A = (/) \/ E. x e. om A = suc x ) )
13.2.11.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 15592*	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 15560 for a bounded version not requiring ax-setind 4573. See finds 4636 for a proof in IZF. From this version, it is easy to prove of finds 4636, finds2 4637, finds1 4638. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( 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-findisg 15593*	Version of bj-findis 15592 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15592 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- F/ x ch   &    |- F/ x th   &    |- ( x = (/) -> ( ps -> ph ) )   &    |- ( x = y -> ( ph -> ch ) )   &    |- ( x = suc y -> ( th -> ph ) )   &    |- F/_ x A   &    |- F/ x ta   &    |- ( x = A -> ( ph -> ta ) )   =>    |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> ( A e. om -> ta ) )
Theorem	bj-findes 15594	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15592 for explanations. From this version, it is easy to prove findes 4639. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )
13.2.12  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 15595*	Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that ph represents a multivalued function on a, or equivalently a collection of nonempty classes indexed by a, and the axiom asserts the existence of a set b which "collects" at least one element in the image of each x e. a and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4148. (Contributed by BJ, 5-Oct-2019.)
|- A. a ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcoll2 15596*	Version of ax-strcoll 15595 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcollnft 15597*	Closed form of strcollnf 15598. (Contributed by BJ, 21-Oct-2019.)
|- ( A. x A. y F/ b ph -> ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) ) )
Theorem	strcollnf 15598*	Version of ax-strcoll 15595 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 15596 with the disjoint variable condition on b , ph replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 15596 will generally suffice: since the theorem asserts the existence of a set b, supposing that that setvar does not occur in the already defined ph is not a big constraint. (Contributed by BJ, 21-Oct-2019.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
Theorem	strcollnfALT 15599*	Alternate proof of strcollnf 15598, not using strcollnft 15597. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ b ph   =>    |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )
13.2.13  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 15600*	Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that ph represents a multivalued function from a to b, or equivalently a collection of nonempty subsets of b indexed by a, and the consequent asserts the existence of a subset of c which "collects" at least one element in the image of each x e. a and which is made only of such elements. The axiom asserts the existence, for any sets a , b, of a set c such that that implication holds for any value of the parameter z of ph. (Contributed by BJ, 5-Oct-2019.)
|- A. a A. b E. c A. z ( A. x e. a E. y e. b ph -> E. d e. c ( A. x e. a E. y e. d ph /\ A. y e. d E. x e. a ph ) )