P. 170 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16901-17000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-axemptylem 16901*	Lemma for bj-axempty 16902 and bj-axempty2 16903. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4257 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
Theorem	bj-axempty 16902*	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 4256. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4257 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥
Theorem	bj-axempty2 16903*	Axiom of the empty set from bounded separation, alternate version to bj-axempty 16902. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4257 instead. (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	bj-nalset 16904*	nalset 4261 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
Theorem	bj-vprc 16905	vprc 4263 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ V
Theorem	bj-nvel 16906	nvel 4264 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ V ∈ 𝐴
Theorem	bj-vnex 16907	vnex 4262 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ ¬ ∃𝑥 𝑥 = V
Theorem	bdinex1 16908	Bounded version of inex1 4265. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∩ 𝐵) ∈ V
Theorem	bdinex2 16909	Bounded version of inex2 4266. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∩ 𝐴) ∈ V
Theorem	bdinex1g 16910	Bounded version of inex1g 4267. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
Theorem	bdssex 16911	Bounded version of ssex 4268. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
Theorem	bdssexi 16912	Bounded version of ssexi 4269. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ V
Theorem	bdssexg 16913	Bounded version of ssexg 4270. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
Theorem	bdssexd 16914	Bounded version of ssexd 4271. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝜑 → 𝐴 ∈ V)
Theorem	bdrabexg 16915*	Bounded version of rabexg 4277. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	bj-inex 16916	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 16917	intexr 4284 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	bj-intnexr 16918	intnexr 4285 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V)
Theorem	bj-zfpair2 16919	Proof of zfpair2 4345 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-prexg 16920	Proof of prexg 4347 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-snexg 16921	snexg 4319 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 16922	snex 4320 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ∈ V
Theorem	bj-sels 16923*	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 16924*	axun2 4578 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 16925*	uniex2 4579 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
⊢ ∃𝑦 𝑦 = ∪ 𝑥
Theorem	bj-uniex 16926	uniex 4581 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ 𝐴 ∈ V
Theorem	bj-uniexg 16927	uniexg 4583 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
Theorem	bj-unex 16928	unex 4585 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ V
Theorem	bdunexb 16929	Bounded version of unexb 4586. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-unexg 16930	unexg 4587 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-sucexg 16931	sucexg 4643 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	bj-sucex 16932	sucex 4644 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ suc 𝐴 ∈ V
14.3.9.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 16933	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 𝜑    ⇒   ⊢ DECID 𝜑
Theorem	bj-d0clsepcl 16934	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 𝜑
14.3.9.2  Inductive classes and the class of natural number ordinals
Syntax	wind 16935	Syntax for inductive classes.
wff Ind 𝐴
Definition	df-bj-ind 16936*	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 16937	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
Theorem	bj-indeq 16938	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
Theorem	bj-bdind 16939	Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED Ind 𝑥
Theorem	bj-indint 16940*	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 16941*	If 𝐴 is inductive and 𝐵 is "inductive in 𝐴 " (a condition weaker than "inductive"), 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 16942	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 16943	ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
⊢ Ind ω
Theorem	bj-omssind 16944	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 16945*	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 16946*	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 16947*	Two formulations of the axiom of infinity (see ax-infvn 16950 and bj-omex 16951) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 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 4739 and peano3 4741 already show this. In this section, we prove bj-peano2 16948 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 16948	Constructive proof of peano2 4740. Temporary note: another possibility is to simply replace sucexg 4643 with bj-sucexg 16931 in the proof of peano2 4740. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Theorem	peano5set 16949*	Version of peano5 4743 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))
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 16950) and deduce that the class ω of natural number ordinals is a set (bj-omex 16951).
Axiom	ax-infvn 16950*	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 4733) from which one then proves, using full separation, that the wanted set exists (omex 4738). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))
Theorem	bj-omex 16951	Proof of omex 4738 from ax-infvn 16950. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
⊢ ω ∈ 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 16952*	Bounded version of peano5 4743. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	speano5 16953*	Version of peano5 4743 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)
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 16954*	Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4744 for a nonconstructive proof of the general case. See bdfind 16955 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 16955*	Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4744 for a nonconstructive proof of the general case. See findset 16954 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 16956*	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 4745 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4745, finds2 4746, finds1 4747. (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 16957*	Version of bj-bdfindis 16956 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 16956 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 16958	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 16956 for explanations. From this version, it is easy to prove the bounded version of findes 4748. (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 16959*	Constructive proof of a variant of nn0suc 4749. For a constructive proof of nn0suc 4749, see bj-nn0suc 16973. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
Theorem	bj-nntrans 16960	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 16961	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Tr 𝐴)
Theorem	bj-nnelirr 16962	A natural number does not belong to itself. Version of elirr 4686 for natural numbers, which does not require ax-setind 4682. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)
Theorem	bj-nnen2lp 16963	A version of en2lp 4699 for natural numbers, which does not require ax-setind 4682. Note: using this theorem and bj-nnelirr 16962, one can remove dependency on ax-setind 4682 from nntri2 6761 and nndcel 6767; 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 16964	Remove from peano4 4742 dependency on ax-setind 4682. 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 16965	The set ω is transitive. A natural number is included in ω. Constructive proof of elomssom 4750. 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 16966	The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Tr ω
Theorem	bj-nnord 16967	A natural number is an ordinal class. Constructive proof of nnord 4757. Can also be proved from bj-nnelon 16968 if the latter is proved from bj-omssonALT 16972. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	bj-nnelon 16968	A natural number is an ordinal. Constructive proof of nnon 4755. Can also be proved from bj-omssonALT 16972. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	bj-omord 16969	The set ω is an ordinal class. Constructive proof of ordom 4752. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Ord ω
Theorem	bj-omelon 16970	The set ω is an ordinal. Constructive proof of omelon 4754. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ ω ∈ On
Theorem	bj-omsson 16971	Constructive proof of omsson 4758. See also bj-omssonALT 16972. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
⊢ ω ⊆ On
Theorem	bj-omssonALT 16972	Alternate proof of bj-omsson 16971. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ⊆ On
Theorem	bj-nn0suc 16973*	Proof of (biconditional form of) nn0suc 4749 from the core axioms of CZF. See also bj-nn0sucALT 16987. As a characterization of the elements of ω, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
14.3.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
14.3.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 16974*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑))
Theorem	setindf 16975*	Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)
Theorem	setindis 16976*	Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑))    ⇒   ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)
Axiom	ax-bdsetind 16977*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
Theorem	bdsetindis 16978*	Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑))    ⇒   ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)
Theorem	bj-inf2vnlem1 16979*	Lemma for bj-inf2vn 16983. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
Theorem	bj-inf2vnlem2 16980*	Lemma for bj-inf2vnlem3 16981 and bj-inf2vnlem4 16982. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))
Theorem	bj-inf2vnlem3 16981*	Lemma for bj-inf2vn 16983. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝑍    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))
Theorem	bj-inf2vnlem4 16982*	Lemma for bj-inf2vn2 16984. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))
Theorem	bj-inf2vn 16983*	A sufficient condition for ω to be a set. See bj-inf2vn2 16984 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
Theorem	bj-inf2vn2 16984*	A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 16983. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
Axiom	ax-inf2 16985*	Another axiom of infinity in a constructive setting (see ax-infvn 16950). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
Theorem	bj-omex2 16986	Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 16950 (see bj-2inf 16947 for the equivalence of the latter with bj-omex 16951). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	bj-nn0sucALT 16987*	Alternate proof of bj-nn0suc 16973, also constructive but from ax-inf2 16985, hence requiring ax-bdsetind 16977. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
14.3.11.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 16988*	Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 16956 for a bounded version not requiring ax-setind 4682. See finds 4745 for a proof in IZF. From this version, it is easy to prove of finds 4745, finds2 4746, finds1 4747. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
⊢ 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-findisg 16989*	Version of bj-findis 16988 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 16988 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ 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-findes 16990	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 16988 for explanations. From this version, it is easy to prove findes 4748. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
14.3.12  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 16991*	Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function on 𝑎, or equivalently a collection of nonempty classes indexed by 𝑎, and the axiom asserts the existence of a set 𝑏 which "collects" at least one element in the image of each 𝑥 ∈ 𝑎 and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4244. (Contributed by BJ, 5-Oct-2019.)
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcoll2 16992*	Version of ax-strcoll 16991 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcollnft 16993*	Closed form of strcollnf 16994. (Contributed by BJ, 21-Oct-2019.)
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
Theorem	strcollnf 16994*	Version of ax-strcoll 16991 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 16992 with the disjoint variable condition on 𝑏, 𝜑 replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 16992 will generally suffice: since the theorem asserts the existence of a set 𝑏, supposing that that setvar does not occur in the already defined 𝜑 is not a big constraint. (Contributed by BJ, 21-Oct-2019.)
⊢ Ⅎ𝑏𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcollnfALT 16995*	Alternate proof of strcollnf 16994, not using strcollnft 16993. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑏𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
14.3.13  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 16996*	Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function from 𝑎 to 𝑏, or equivalently a collection of nonempty subsets of 𝑏 indexed by 𝑎, and the consequent asserts the existence of a subset of 𝑐 which "collects" at least one element in the image of each 𝑥 ∈ 𝑎 and which is made only of such elements. The axiom asserts the existence, for any sets 𝑎, 𝑏, of a set 𝑐 such that that implication holds for any value of the parameter 𝑧 of 𝜑. (Contributed by BJ, 5-Oct-2019.)
⊢ ∀𝑎∀𝑏∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))
Theorem	sscoll2 16997*	Version of ax-sscoll 16996 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑))
14.3.14  Real numbers
Axiom	ax-ddkcomp 16998	Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then Axiom ax-ddkcomp 16998 should be used in place of construction specific results. In particular, axcaucvg 8261 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝐵) → 𝑥 ≤ 𝐵)))
14.4  Mathbox for Jim Kingdon
14.4.1  Propositional and predicate logic
Theorem	nnnotnotr 16999	Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 862, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
⊢ ¬ ¬ (¬ ¬ 𝜑 → 𝜑)
14.4.2  The sizes of sets
Theorem	ss1oel2o 17000	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4333 which more directly illustrates the contrast with el2oss1o 6710. (Contributed by Jim Kingdon, 8-Aug-2022.)
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o))