P. 165 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16401-16500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdcsn 16401	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x }
Theorem	bdcpr 16402	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y }
Theorem	bdctp 16403	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y , z }
Theorem	bdsnss 16404*	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 16405*	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 16406	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <. x , y >.
Theorem	bdcuni 16407	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U. x
Theorem	bdcint 16408	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^| x
Theorem	bdciun 16409*	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 16410*	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 16411	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 16412*	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 16413	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 16414*	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 )
14.2.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 16415*	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 4205. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep1 16416*	Version of ax-bdsep 16415 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep2 16417*	Version of ax-bdsep 16415 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 16416 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnft 16418*	Closed form of bdsepnf 16419. Version of ax-bdsep 16415 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 16416 when sufficient. (Contributed by BJ, 19-Oct-2019.)
|- BOUNDED ph   =>    |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )
Theorem	bdsepnf 16419*	Version of ax-bdsep 16415 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 16420. Use bdsep1 16416 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 16420*	Alternate proof of bdsepnf 16419, not using bdsepnft 16418. (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 16421*	Closed form of the version of zfauscl 4207 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 16422*	Bounded version of bm1.3ii 4208. (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 16423*	Lemma for bj-axempty 16424 and bj-axempty2 16425. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4213 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 16424*	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 4212. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4213 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 16425*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 16424. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4213 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 16426*	nalset 4217 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x
Theorem	bj-vprc 16427	vprc 4219 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 16428	nvel 4220 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 16429	vnex 4218 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 16430	Bounded version of inex1 4221. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 16431	Bounded version of inex2 4222. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 16432	Bounded version of inex1g 4223. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 16433	Bounded version of ssex 4224. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 16434	Bounded version of ssexi 4225. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	bdssexg 16435	Bounded version of ssexg 4226. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	bdssexd 16436	Bounded version of ssexd 4227. (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 16437*	Bounded version of rabexg 4231. (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 16438	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 16439	intexr 4238 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	bj-intnexr 16440	intnexr 4239 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A = _V -> -. |^| A e. _V )
Theorem	bj-zfpair2 16441	Proof of zfpair2 4298 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- { x , y } e. _V
Theorem	bj-prexg 16442	Proof of prexg 4299 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 16443	snexg 4272 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ( A e. V -> { A } e. _V )
Theorem	bj-snex 16444	snex 4273 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- { A } e. _V
Theorem	bj-sels 16445*	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 16446*	axun2 4530 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 16447*	uniex2 4531 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y y = U. x
Theorem	bj-uniex 16448	uniex 4532 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- U. A e. _V
Theorem	bj-uniexg 16449	uniexg 4534 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> U. A e. _V )
Theorem	bj-unex 16450	unex 4536 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 16451	Bounded version of unexb 4537. (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 16452	unexg 4538 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 16453	sucexg 4594 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> suc A e. _V )
Theorem	bj-sucex 16454	sucex 4595 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- suc A e. _V
14.2.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 16455	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
|- BOUNDED ph   =>    |- DECID ph
Theorem	bj-d0clsepcl 16456	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
|- DECID ph
14.2.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 16457	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 16458*	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 16459	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 16460	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- ( A = B -> (Ind A <-> Ind B ) )
Theorem	bj-bdind 16461	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 16462*	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 16463*	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 16464	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 16465	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 16466	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 16467*	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 16468*	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 16469*	Two formulations of the axiom of infinity (see ax-infvn 16472 and bj-omex 16473) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E. x (Ind x /\ A. y (Ind y -> x C_ y ) ) )
14.2.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4690 and peano3 4692 already show this. In this section, we prove bj-peano2 16470 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 16470	Constructive proof of peano2 4691. Temporary note: another possibility is to simply replace sucexg 4594 with bj-sucexg 16453 in the proof of peano2 4691. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> suc A e. om )
Theorem	peano5set 16471*	Version of peano5 4694 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 ) )
14.2.10  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.
14.2.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 16472) and deduce that the class om of natural number ordinals is a set (bj-omex 16473).
Axiom	ax-infvn 16472*	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 4684) from which one then proves, using full separation, that the wanted set exists (omex 4689). "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 16473	Proof of omex 4689 from ax-infvn 16472. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
14.2.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 16474*	Bounded version of peano5 4694. (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 16475*	Version of peano5 4694 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 )
14.2.10.3  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 16476*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4695 for a nonconstructive proof of the general case. See bdfind 16477 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 16477*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4695 for a nonconstructive proof of the general case. See findset 16476 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 16478*	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 4696 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4696, finds2 4697, finds1 4698. (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 16479*	Version of bj-bdfindis 16478 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16478 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 16480	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 16478 for explanations. From this version, it is easy to prove the bounded version of findes 4699. (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 16481*	Constructive proof of a variant of nn0suc 4700. For a constructive proof of nn0suc 4700, see bj-nn0suc 16495. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 16482	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 16483	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 16484	A natural number does not belong to itself. Version of elirr 4637 for natural numbers, which does not require ax-setind 4633. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 16485	A version of en2lp 4650 for natural numbers, which does not require ax-setind 4633. Note: using this theorem and bj-nnelirr 16484, one can remove dependency on ax-setind 4633 from nntri2 6657 and nndcel 6663; 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 16486	Remove from peano4 4693 dependency on ax-setind 4633. 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 16487	The set om is transitive. A natural number is included in om. Constructive proof of elnn 4702. 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 16488	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om
Theorem	bj-nnord 16489	A natural number is an ordinal class. Constructive proof of nnord 4708. Can also be proved from bj-nnelon 16490 if the latter is proved from bj-omssonALT 16494. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> Ord A )
Theorem	bj-nnelon 16490	A natural number is an ordinal. Constructive proof of nnon 4706. Can also be proved from bj-omssonALT 16494. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
|- ( A e. om -> A e. On )
Theorem	bj-omord 16491	The set om is an ordinal class. Constructive proof of ordom 4703. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Ord om
Theorem	bj-omelon 16492	The set om is an ordinal. Constructive proof of omelon 4705. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- om e. On
Theorem	bj-omsson 16493	Constructive proof of omsson 4709. See also bj-omssonALT 16494. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
|- om C_ On
Theorem	bj-omssonALT 16494	Alternate proof of bj-omsson 16493. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
|- om C_ On
Theorem	bj-nn0suc 16495*	Proof of (biconditional form of) nn0suc 4700 from the core axioms of CZF. See also bj-nn0sucALT 16509. 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 ) )
14.2.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
14.2.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 16496*	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 16497*	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 16498*	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 16499*	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 16500*	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 )