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	bdel 15501*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdeli 15502*	Inference associated with bdel 15501. Its converse is bdelir 15503. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∈ 𝐴
Theorem	bdelir 15503*	Inference associated with df-bdc 15497. Its converse is bdeli 15502. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝐴    ⇒   ⊢ BOUNDED 𝐴
Theorem	bdcv 15504	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥
Theorem	bdcab 15505	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∣ 𝜑}
Theorem	bdph 15506	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED {𝑥 ∣ 𝜑}    ⇒   ⊢ BOUNDED 𝜑
Theorem	bds 15507*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 15478; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 15478. (Contributed by BJ, 19-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdcrab 15508*	A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bdne 15509	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 ≠ 𝑦
Theorem	bdnel 15510*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∉ 𝐴
Theorem	bdreu 15511*	Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 15513, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 15480, if ∀𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then ∀𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
Theorem	bdrmo 15512*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑
Theorem	bdcvv 15513	The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED V
Theorem	bdsbc 15514	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 15515. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdsbcALT 15515	Alternate proof of bdsbc 15514. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdccsb 15516	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 15517	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∖ 𝐵)
Theorem	bdcun 15518	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∪ 𝐵)
Theorem	bdcin 15519	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∩ 𝐵)
Theorem	bdss 15520	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 ⊆ 𝐴
Theorem	bdcnul 15521	The empty class is bounded. See also bdcnulALT 15522. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ∅
Theorem	bdcnulALT 15522	Alternate proof of bdcnul 15521. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 15500, or use the corresponding characterizations of its elements followed by bdelir 15503. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED ∅
Theorem	bdeq0 15523	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 15524	Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED ∅ ∈ 𝑥
Theorem	bdcpw 15525	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝒫 𝐴
Theorem	bdcsn 15526	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥}
Theorem	bdcpr 15527	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦}
Theorem	bdctp 15528	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦, 𝑧}
Theorem	bdsnss 15529*	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} ⊆ 𝐴
Theorem	bdvsn 15530*	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 15531	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED ⟨𝑥, 𝑦⟩
Theorem	bdcuni 15532	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
⊢ BOUNDED ∪ 𝑥
Theorem	bdcint 15533	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED ∩ 𝑥
Theorem	bdciun 15534*	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 ∪ 𝑥 ∈ 𝑦 𝐴
Theorem	bdciin 15535*	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 ∈ 𝑦 𝐴
Theorem	bdcsuc 15536	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED suc 𝑥
Theorem	bdeqsuc 15537*	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 15538	Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED suc 𝑥 ∈ 𝑦
Theorem	bdcriota 15539*	A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ ∃!𝑥 ∈ 𝑦 𝜑    ⇒   ⊢ BOUNDED (℩𝑥 ∈ 𝑦 𝜑)
13.2.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 15540*	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 4152. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep1 15541*	Version of ax-bdsep 15540 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep2 15542*	Version of ax-bdsep 15540 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 15541 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnft 15543*	Closed form of bdsepnf 15544. Version of ax-bdsep 15540 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 15541 when sufficient. (Contributed by BJ, 19-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
Theorem	bdsepnf 15544*	Version of ax-bdsep 15540 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 15545. Use bdsep1 15541 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ Ⅎ𝑏𝜑    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnfALT 15545*	Alternate proof of bdsepnf 15544, not using bdsepnft 15543. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑏𝜑    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdzfauscl 15546*	Closed form of the version of zfauscl 4154 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	bdbm1.3ii 15547*	Bounded version of bm1.3ii 4155. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)    ⇒   ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
Theorem	bj-axemptylem 15548*	Lemma for bj-axempty 15549 and bj-axempty2 15550. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4160 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
Theorem	bj-axempty 15549*	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 4159. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4160 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥
Theorem	bj-axempty2 15550*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 15549. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4160 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	bj-nalset 15551*	nalset 4164 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	bj-vprc 15552	vprc 4166 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ V
Theorem	bj-nvel 15553	nvel 4167 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	bj-vnex 15554	vnex 4165 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	bdinex1 15555	Bounded version of inex1 4168. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ V
Theorem	bdinex2 15556	Bounded version of inex2 4169. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	bdinex1g 15557	Bounded version of inex1g 4170. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bdssex 15558	Bounded version of ssex 4171. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	bdssexi 15559	Bounded version of ssexi 4172. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	bdssexg 15560	Bounded version of ssexg 4173. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	bdssexd 15561	Bounded version of ssexd 4174. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	bdrabexg 15562*	Bounded version of rabexg 4177. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	bj-inex 15563	The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bj-intexr 15564	intexr 4184 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	bj-intnexr 15565	intnexr 4185 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
Theorem	bj-zfpair2 15566	Proof of zfpair2 4244 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-prexg 15567	Proof of prexg 4245 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-snexg 15568	snexg 4218 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 15569	snex 4219 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ∈ V
Theorem	bj-sels 15570*	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 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	bj-axun2 15571*	axun2 4471 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
Theorem	bj-uniex2 15572*	uniex2 4472 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
⊢ ∃𝑦 𝑦 = ∪ 𝑥
Theorem	bj-uniex 15573	uniex 4473 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ 𝐴 ∈ V
Theorem	bj-uniexg 15574	uniexg 4475 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
Theorem	bj-unex 15575	unex 4477 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ V
Theorem	bdunexb 15576	Bounded version of unexb 4478. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-unexg 15577	unexg 4479 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-sucexg 15578	sucexg 4535 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	bj-sucex 15579	sucex 4536 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ suc 𝐴 ∈ V
13.2.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 15580	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
⊢ BOUNDED 𝜑    ⇒   ⊢ DECID 𝜑
Theorem	bj-d0clsepcl 15581	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
⊢ DECID 𝜑
13.2.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 15582	Syntax for inductive classes.
wff Ind 𝐴
Definition	df-bj-ind 15583*	Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
Theorem	bj-indsuc 15584	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
Theorem	bj-indeq 15585	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
Theorem	bj-bdind 15586	Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED Ind 𝑥
Theorem	bj-indint 15587*	The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
Theorem	bj-indind 15588*	If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴 ∩ 𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵))
Theorem	bj-dfom 15589	Alternate definition of ω, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
⊢ ω = ∩ {𝑥 ∣ Ind 𝑥}
Theorem	bj-omind 15590	ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
⊢ Ind ω
Theorem	bj-omssind 15591	ω 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 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
Theorem	bj-ssom 15592*	A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω)
Theorem	bj-om 15593*	A set is equal to ω if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))))
Theorem	bj-2inf 15594*	Two formulations of the axiom of infinity (see ax-infvn 15597 and bj-omex 15598) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 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 4631 and peano3 4633 already show this. In this section, we prove bj-peano2 15595 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 15595	Constructive proof of peano2 4632. Temporary note: another possibility is to simply replace sucexg 4535 with bj-sucexg 15578 in the proof of peano2 4632. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Theorem	peano5set 15596*	Version of peano5 4635 when ω ∩ 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.)
⊢ ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 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 15597) and deduce that the class ω of natural number ordinals is a set (bj-omex 15598).
Axiom	ax-infvn 15597*	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 4625) from which one then proves, using full separation, that the wanted set exists (omex 4630). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))
Theorem	bj-omex 15598	Proof of omex 4630 from ax-infvn 15597. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
⊢ ω ∈ 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 15599*	Bounded version of peano5 4635. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	speano5 15600*	Version of peano5 4635 when 𝐴 is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)