P. 150 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14901-15000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bdne 14901	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 ≠ 𝑦
Theorem	bdnel 14902*	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 14903*	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 14905, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 14872, 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 14904*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑
Theorem	bdcvv 14905	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 14906	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 14907. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdsbcALT 14907	Alternate proof of bdsbc 14906. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdccsb 14908	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 14909	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∖ 𝐵)
Theorem	bdcun 14910	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∪ 𝐵)
Theorem	bdcin 14911	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∩ 𝐵)
Theorem	bdss 14912	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 14913	The empty class is bounded. See also bdcnulALT 14914. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ∅
Theorem	bdcnulALT 14914	Alternate proof of bdcnul 14913. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 14892, or use the corresponding characterizations of its elements followed by bdelir 14895. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED ∅
Theorem	bdeq0 14915	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 14916	Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED ∅ ∈ 𝑥
Theorem	bdcpw 14917	The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝒫 𝐴
Theorem	bdcsn 14918	The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥}
Theorem	bdcpr 14919	The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦}
Theorem	bdctp 14920	The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED {𝑥, 𝑦, 𝑧}
Theorem	bdsnss 14921*	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 14922*	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 14923	The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
⊢ BOUNDED ⟨𝑥, 𝑦⟩
Theorem	bdcuni 14924	The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
⊢ BOUNDED ∪ 𝑥
Theorem	bdcint 14925	The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED ∩ 𝑥
Theorem	bdciun 14926*	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 14927*	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 14928	The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED suc 𝑥
Theorem	bdeqsuc 14929*	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 14930	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 14931*	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 14932*	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 4133. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep1 14933*	Version of ax-bdsep 14932 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsep2 14934*	Version of ax-bdsep 14932 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 14933 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnft 14935*	Closed form of bdsepnf 14936. Version of ax-bdsep 14932 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 14933 when sufficient. (Contributed by BJ, 19-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
Theorem	bdsepnf 14936*	Version of ax-bdsep 14932 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 14937. Use bdsep1 14933 when sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ Ⅎ𝑏𝜑    &   ⊢ BOUNDED 𝜑    ⇒   ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
Theorem	bdsepnfALT 14937*	Alternate proof of bdsepnf 14936, not using bdsepnft 14935. (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 14938*	Closed form of the version of zfauscl 4135 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 14939*	Bounded version of bm1.3ii 4136. (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 14940*	Lemma for bj-axempty 14941 and bj-axempty2 14942. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
Theorem	bj-axempty 14941*	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 4140. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥
Theorem	bj-axempty2 14942*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 14941. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	bj-nalset 14943*	nalset 4145 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	bj-vprc 14944	vprc 4147 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ V
Theorem	bj-nvel 14945	nvel 4148 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	bj-vnex 14946	vnex 4146 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	bdinex1 14947	Bounded version of inex1 4149. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ V
Theorem	bdinex2 14948	Bounded version of inex2 4150. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	bdinex1g 14949	Bounded version of inex1g 4151. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bdssex 14950	Bounded version of ssex 4152. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	bdssexi 14951	Bounded version of ssexi 4153. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	bdssexg 14952	Bounded version of ssexg 4154. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	bdssexd 14953	Bounded version of ssexd 4155. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	bdrabexg 14954*	Bounded version of rabexg 4158. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	bj-inex 14955	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 14956	intexr 4162 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	bj-intnexr 14957	intnexr 4163 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
Theorem	bj-zfpair2 14958	Proof of zfpair2 4222 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-prexg 14959	Proof of prexg 4223 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-snexg 14960	snexg 4196 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 14961	snex 4197 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ∈ V
Theorem	bj-sels 14962*	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 14963*	axun2 4447 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 14964*	uniex2 4448 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
⊢ ∃𝑦 𝑦 = ∪ 𝑥
Theorem	bj-uniex 14965	uniex 4449 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ 𝐴 ∈ V
Theorem	bj-uniexg 14966	uniexg 4451 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
Theorem	bj-unex 14967	unex 4453 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ V
Theorem	bdunexb 14968	Bounded version of unexb 4454. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-unexg 14969	unexg 4455 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-sucexg 14970	sucexg 4509 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	bj-sucex 14971	sucex 4510 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 14972	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
⊢ BOUNDED 𝜑    ⇒   ⊢ DECID 𝜑
Theorem	bj-d0clsepcl 14973	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 14974	Syntax for inductive classes.
wff Ind 𝐴
Definition	df-bj-ind 14975*	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 14976	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
Theorem	bj-indeq 14977	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
Theorem	bj-bdind 14978	Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED Ind 𝑥
Theorem	bj-indint 14979*	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 14980*	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 14981	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 14982	ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
⊢ Ind ω
Theorem	bj-omssind 14983	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 14984*	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 14985*	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 14986*	Two formulations of the axiom of infinity (see ax-infvn 14989 and bj-omex 14990) . (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 4605 and peano3 4607 already show this. In this section, we prove bj-peano2 14987 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 14987	Constructive proof of peano2 4606. Temporary note: another possibility is to simply replace sucexg 4509 with bj-sucexg 14970 in the proof of peano2 4606. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Theorem	peano5set 14988*	Version of peano5 4609 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 14989) and deduce that the class ω of natural number ordinals is a set (bj-omex 14990).
Axiom	ax-infvn 14989*	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 4599) from which one then proves, using full separation, that the wanted set exists (omex 4604). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))
Theorem	bj-omex 14990	Proof of omex 4604 from ax-infvn 14989. (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 14991*	Bounded version of peano5 4609. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	speano5 14992*	Version of peano5 4609 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 14993*	Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4610 for a nonconstructive proof of the general case. See bdfind 14994 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 14994*	Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4610 for a nonconstructive proof of the general case. See findset 14993 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 14995*	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 4611 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4611, finds2 4612, finds1 4613. (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 14996*	Version of bj-bdfindis 14995 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 14995 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 14997	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 14995 for explanations. From this version, it is easy to prove the bounded version of findes 4614. (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 14998*	Constructive proof of a variant of nn0suc 4615. For a constructive proof of nn0suc 4615, see bj-nn0suc 15012. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
Theorem	bj-nntrans 14999	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 15000	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Tr 𝐴)