P. 148 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14701-14800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdss 14701	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 14702	The empty class is bounded. See also bdcnulALT 14703. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ∅
Theorem	bdcnulALT 14703	Alternate proof of bdcnul 14702. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 14681, or use the corresponding characterizations of its elements followed by bdelir 14684. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED ∅
Theorem	bdeq0 14704	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 14705	Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED ∅ ∈ 𝑥
Theorem	bdcpw 14706	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝒫 𝐴
Theorem	bdcsn 14707	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥}
Theorem	bdcpr 14708	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦}
Theorem	bdctp 14709	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦, 𝑧}
Theorem	bdsnss 14710*	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 14711*	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 14712	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED ⟨𝑥, 𝑦⟩
Theorem	bdcuni 14713	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
⊢ BOUNDED ∪ 𝑥
Theorem	bdcint 14714	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED ∩ 𝑥
Theorem	bdciun 14715*	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 14716*	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 14717	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED suc 𝑥
Theorem	bdeqsuc 14718*	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 14719	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 14720*	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)
12.2.9  CZF: Bounded separation In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.
Axiom	ax-bdsep 14721*	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 4123. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep1 14722*	Version of ax-bdsep 14721 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep2 14723*	Version of ax-bdsep 14721 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 14722 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnft 14724*	Closed form of bdsepnf 14725. Version of ax-bdsep 14721 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 14722 when sufficient. (Contributed by BJ, 19-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
Theorem	bdsepnf 14725*	Version of ax-bdsep 14721 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 14726. Use bdsep1 14722 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ Ⅎ𝑏𝜑    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnfALT 14726*	Alternate proof of bdsepnf 14725, not using bdsepnft 14724. (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 14727*	Closed form of the version of zfauscl 4125 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 14728*	Bounded version of bm1.3ii 4126. (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 14729*	Lemma for bj-axempty 14730 and bj-axempty2 14731. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4131 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
Theorem	bj-axempty 14730*	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 4130. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4131 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥
Theorem	bj-axempty2 14731*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 14730. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4131 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	bj-nalset 14732*	nalset 4135 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	bj-vprc 14733	vprc 4137 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ V
Theorem	bj-nvel 14734	nvel 4138 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	bj-vnex 14735	vnex 4136 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	bdinex1 14736	Bounded version of inex1 4139. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ V
Theorem	bdinex2 14737	Bounded version of inex2 4140. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	bdinex1g 14738	Bounded version of inex1g 4141. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bdssex 14739	Bounded version of ssex 4142. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	bdssexi 14740	Bounded version of ssexi 4143. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	bdssexg 14741	Bounded version of ssexg 4144. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	bdssexd 14742	Bounded version of ssexd 4145. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	bdrabexg 14743*	Bounded version of rabexg 4148. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	bj-inex 14744	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 14745	intexr 4152 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	bj-intnexr 14746	intnexr 4153 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
Theorem	bj-zfpair2 14747	Proof of zfpair2 4212 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-prexg 14748	Proof of prexg 4213 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-snexg 14749	snexg 4186 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 14750	snex 4187 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ∈ V
Theorem	bj-sels 14751*	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 14752*	axun2 4437 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 14753*	uniex2 4438 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
⊢ ∃𝑦 𝑦 = ∪ 𝑥
Theorem	bj-uniex 14754	uniex 4439 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ 𝐴 ∈ V
Theorem	bj-uniexg 14755	uniexg 4441 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
Theorem	bj-unex 14756	unex 4443 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ V
Theorem	bdunexb 14757	Bounded version of unexb 4444. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-unexg 14758	unexg 4445 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-sucexg 14759	sucexg 4499 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	bj-sucex 14760	sucex 4500 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ suc 𝐴 ∈ V
12.2.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 14761	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
⊢ BOUNDED 𝜑    ⇒   ⊢ DECID 𝜑
Theorem	bj-d0clsepcl 14762	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
⊢ DECID 𝜑
12.2.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 14763	Syntax for inductive classes.
wff Ind 𝐴
Definition	df-bj-ind 14764*	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 14765	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
Theorem	bj-indeq 14766	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
Theorem	bj-bdind 14767	Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED Ind 𝑥
Theorem	bj-indint 14768*	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 14769*	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 14770	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 14771	ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
⊢ Ind ω
Theorem	bj-omssind 14772	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 14773*	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 14774*	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 14775*	Two formulations of the axiom of infinity (see ax-infvn 14778 and bj-omex 14779) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))
12.2.9.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4595 and peano3 4597 already show this. In this section, we prove bj-peano2 14776 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 14776	Constructive proof of peano2 4596. Temporary note: another possibility is to simply replace sucexg 4499 with bj-sucexg 14759 in the proof of peano2 4596. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Theorem	peano5set 14777*	Version of peano5 4599 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))
12.2.10  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.
12.2.10.1  The set of natural number ordinals In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 14778) and deduce that the class ω of natural number ordinals is a set (bj-omex 14779).
Axiom	ax-infvn 14778*	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 4589) from which one then proves, using full separation, that the wanted set exists (omex 4594). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))
Theorem	bj-omex 14779	Proof of omex 4594 from ax-infvn 14778. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
⊢ ω ∈ V
12.2.10.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 14780*	Bounded version of peano5 4599. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	speano5 14781*	Version of peano5 4599 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)
12.2.10.3  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 14782*	Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4600 for a nonconstructive proof of the general case. See bdfind 14783 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω))
Theorem	bdfind 14783*	Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4600 for a nonconstructive proof of the general case. See findset 14782 for a proof when 𝐴 is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)
Theorem	bj-bdfindis 14784*	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 4601 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4601, finds2 4602, finds1 4603. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑥𝜃    &   ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))    ⇒   ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
Theorem	bj-bdfindisg 14785*	Version of bj-bdfindis 14784 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 14784 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑥𝜃    &   ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜏    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏))    ⇒   ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏))
Theorem	bj-bdfindes 14786	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 14784 for explanations. From this version, it is easy to prove the bounded version of findes 4604. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
Theorem	bj-nn0suc0 14787*	Constructive proof of a variant of nn0suc 4605. For a constructive proof of nn0suc 4605, see bj-nn0suc 14801. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
Theorem	bj-nntrans 14788	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	bj-nntrans2 14789	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Tr 𝐴)
Theorem	bj-nnelirr 14790	A natural number does not belong to itself. Version of elirr 4542 for natural numbers, which does not require ax-setind 4538. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)
Theorem	bj-nnen2lp 14791	A version of en2lp 4555 for natural numbers, which does not require ax-setind 4538. Note: using this theorem and bj-nnelirr 14790, one can remove dependency on ax-setind 4538 from nntri2 6497 and nndcel 6503; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
Theorem	bj-peano4 14792	Remove from peano4 4598 dependency on ax-setind 4538. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-omtrans 14793	The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4607. The idea is to use bounded induction with the formula 𝑥 ⊆ ω. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with 𝑥 ⊆ 𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
Theorem	bj-omtrans2 14794	The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Tr ω
Theorem	bj-nnord 14795	A natural number is an ordinal class. Constructive proof of nnord 4613. Can also be proved from bj-nnelon 14796 if the latter is proved from bj-omssonALT 14800. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	bj-nnelon 14796	A natural number is an ordinal. Constructive proof of nnon 4611. Can also be proved from bj-omssonALT 14800. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	bj-omord 14797	The set ω is an ordinal class. Constructive proof of ordom 4608. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Ord ω
Theorem	bj-omelon 14798	The set ω is an ordinal. Constructive proof of omelon 4610. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ ω ∈ On
Theorem	bj-omsson 14799	Constructive proof of omsson 4614. See also bj-omssonALT 14800. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
⊢ ω ⊆ On
Theorem	bj-omssonALT 14800	Alternate proof of bj-omsson 14799. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ⊆ On