P. 172 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 17101-17200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdsbc 17101	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 17102. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdsbcALT 17102	Alternate proof of bdsbc 17101. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdccsb 17103	A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED [_ y / x ]_ A
Theorem	bdcdif 17104	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A \ B )
Theorem	bdcun 17105	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A u. B )
Theorem	bdcin 17106	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A i^i B )
Theorem	bdss 17107	The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x C_ A
Theorem	bdcnul 17108	The empty class is bounded. See also bdcnulALT 17109. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED (/)
Theorem	bdcnulALT 17109	Alternate proof of bdcnul 17108. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 17087, or use the corresponding characterizations of its elements followed by bdelir 17090. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED (/)
Theorem	bdeq0 17110	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 17111	Boundedness of the formula "the empty set belongs to the setvar x". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED (/) e. x
Theorem	bdcpw 17112	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED ~P A
Theorem	bdcsn 17113	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x }
Theorem	bdcpr 17114	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y }
Theorem	bdctp 17115	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED { x , y , z }
Theorem	bdsnss 17116*	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 17117*	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 17118	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
|- BOUNDED <. x , y >.
Theorem	bdcuni 17119	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
|- BOUNDED U. x
Theorem	bdcint 17120	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED |^| x
Theorem	bdciun 17121*	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 17122*	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 17123	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED suc x
Theorem	bdeqsuc 17124*	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 17125	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 17126*	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 )
14.3.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 17127*	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 4251. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsep1 17128*	Version of ax-bdsep 17127 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 17129*	Version of ax-bdsep 17127 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 17128 when sufficient. (Contributed by BJ, 5-Oct-2019.)
|- BOUNDED ph   =>    |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
Theorem	bdsepnft 17130*	Closed form of bdsepnf 17131. Version of ax-bdsep 17127 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 17128 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 17131*	Version of ax-bdsep 17127 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 17132. Use bdsep1 17128 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 17132*	Alternate proof of bdsepnf 17131, not using bdsepnft 17130. (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	bdsepg 17133*	Version of sepg 4253 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 17134*	Bounded version of bm1.3ii 4256. (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 17135*	Lemma for bj-axempty 17136 and bj-axempty2 17137. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4261 instead. (New usage is discouraged.)
|- E. x A. y ( y e. x -> F. )
Theorem	bj-axempty 17136*	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 4260. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4261 instead. (New usage is discouraged.)
|- E. x A. y e. x F.
Theorem	bj-axempty2 17137*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 17136. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4261 instead. (New usage is discouraged.)
|- E. x A. y -. y e. x
Theorem	bj-nalset 17138*	nalset 4265 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x A. y y e. x
Theorem	bj-vprc 17139	vprc 4267 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. _V
Theorem	bj-nvel 17140	nvel 4268 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. _V e. A
Theorem	bj-vnex 17141	vnex 4266 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- -. E. x x = _V
Theorem	bdinex1 17142	Bounded version of inex1 4269. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( A i^i B ) e. _V
Theorem	bdinex2 17143	Bounded version of inex2 4270. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   &    |- A e. _V   =>    |- ( B i^i A ) e. _V
Theorem	bdinex1g 17144	Bounded version of inex1g 4271. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED B   =>    |- ( A e. V -> ( A i^i B ) e. _V )
Theorem	bdssex 17145	Bounded version of ssex 4272. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   =>    |- ( A C_ B -> A e. _V )
Theorem	bdssexi 17146	Bounded version of ssexi 4273. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   &    |- B e. _V   &    |- A C_ B   =>    |- A e. _V
Theorem	bdssexg 17147	Bounded version of ssexg 4274. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- BOUNDED A   =>    |- ( ( A C_ B /\ B e. C ) -> A e. _V )
Theorem	bdssexd 17148	Bounded version of ssexd 4275. (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 17149*	Bounded version of rabexg 4281. (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 17150	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 17151	intexr 4288 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A e. _V -> A =/= (/) )
Theorem	bj-intnexr 17152	intnexr 4289 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( |^| A = _V -> -. |^| A e. _V )
Theorem	bj-zfpair2 17153	Proof of zfpair2 4349 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- { x , y } e. _V
Theorem	bj-prexg 17154	Proof of prexg 4351 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 17155	snexg 4323 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- ( A e. V -> { A } e. _V )
Theorem	bj-snex 17156	snex 4324 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- { A } e. _V
Theorem	bj-sels 17157*	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 17158*	axun2 4582 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 17159*	uniex2 4583 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
|- E. y y = U. x
Theorem	bj-uniex 17160	uniex 4585 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- U. A e. _V
Theorem	bj-uniexg 17161	uniexg 4587 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> U. A e. _V )
Theorem	bj-unex 17162	unex 4589 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 17163	Bounded version of unexb 4590. (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 17164	unexg 4591 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 17165	sucexg 4647 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. V -> suc A e. _V )
Theorem	bj-sucex 17166	sucex 4648 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
|- A e. _V   =>    |- suc A e. _V
14.3.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 17167	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.) New usage is discouraged since this statement is not intuitionnistic. (New usage is discouraged.)
|- BOUNDED ph   =>    |- DECID ph
Theorem	bj-d0clsepcl 17168	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) New usage is discouraged since this statement is not intuitionnistic. (New usage is discouraged.)
|- DECID ph
14.3.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 17169	Syntax for inductive classes.
wff Ind A
Definition	df-bj-ind 17170*	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 17171	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 17172	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
|- ( A = B -> (Ind A <-> Ind B ) )
Theorem	bj-bdind 17173	Boundedness of the formula "the setvar x is an inductive class". (Contributed by BJ, 30-Nov-2019.)
|- BOUNDED Ind x
Theorem	bj-indint 17174*	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 17175*	If A is inductive and B is "inductive in A " (a condition weaker than "inductive"), 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 17176	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 17177	om is an inductive class. (Contributed by BJ, 30-Nov-2019.)
|- Ind om
Theorem	bj-omssind 17178	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 17179*	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 17180*	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 17181*	Two formulations of the axiom of infinity (see ax-infvn 17184 and bj-omex 17185) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
|- ( om e. _V <-> E. x (Ind x /\ A. y (Ind y -> x C_ y ) ) )
14.3.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4743 and peano3 4745 already show this. In this section, we prove bj-peano2 17182 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 17182	Constructive proof of peano2 4744. Temporary note: another possibility is to simply replace sucexg 4647 with bj-sucexg 17165 in the proof of peano2 4744. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> suc A e. om )
Theorem	peano5set 17183*	Version of peano5 4747 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 ) )
14.3.10  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.
14.3.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 17184) and deduce that the class om of natural number ordinals is a set (bj-omex 17185).
Axiom	ax-infvn 17184*	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 4737) from which one then proves, using full separation, that the wanted set exists (omex 4742). "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 17185	Proof of omex 4742 from ax-infvn 17184. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
|- om e. _V
14.3.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 17186*	Bounded version of peano5 4747. (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 17187*	Version of peano5 4747 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 )
14.3.10.3  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 17188*	Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4748 for a nonconstructive proof of the general case. See bdfind 17189 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 17189*	Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4748 for a nonconstructive proof of the general case. See findset 17188 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 17190*	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 4749 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4749, finds2 4750, finds1 4751. (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 17191*	Version of bj-bdfindis 17190 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 17190 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 17192	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 17190 for explanations. From this version, it is easy to prove the bounded version of findes 4752. (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 17193*	Constructive proof of a variant of nn0suc 4753. For a constructive proof of nn0suc 4753, see bj-nn0suc 17207. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> ( A = (/) \/ E. x e. A A = suc x ) )
Theorem	bj-nntrans 17194	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 17195	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 17196	A natural number does not belong to itself. Version of elirr 4690 for natural numbers, which does not require ax-setind 4686. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
|- ( A e. om -> -. A e. A )
Theorem	bj-nnen2lp 17197	A version of en2lp 4703 for natural numbers, which does not require ax-setind 4686. Note: using this theorem and bj-nnelirr 17196, one can remove dependency on ax-setind 4686 from nntri2 6771 and nndcel 6777; 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 17198	Remove from peano4 4746 dependency on ax-setind 4686. 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 17199	The set om is transitive. A natural number is included in om. Constructive proof of elomssom 4754. 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 17200	The set om is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
|- Tr om