P. 108 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10701-10795   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-zfpair2 10701	Proof of zfpair2 3965 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	bj-prexg 10702	Proof of prexg 3966 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	bj-snexg 10703	snexg 3956 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	bj-snex 10704	snex 3957 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ∈ V
Theorem	bj-sels 10705*	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 10706*	axun2 4190 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 10707*	uniex2 4191 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
⊢ ∃𝑦 𝑦 = ∪ 𝑥
Theorem	bj-uniex 10708	uniex 4192 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ 𝐴 ∈ V
Theorem	bj-uniexg 10709	uniexg 4193 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
Theorem	bj-unex 10710	unex 4194 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ V
Theorem	bdunexb 10711	Bounded version of unexb 4195. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-unexg 10712	unexg 4196 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
Theorem	bj-sucexg 10713	sucexg 4242 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
Theorem	bj-sucex 10714	sucex 4243 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ suc 𝐴 ∈ V
6.2.5.1  Delta_0-classical logic
Axiom	ax-bj-d0cl 10715	Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
⊢ BOUNDED 𝜑    ⇒   ⊢ DECID 𝜑
Theorem	bj-notbi 10716	Equivalence property for negation. TODO: minimize all theorems using notbid 624 and notbii 626. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	bj-notbii 10717	Inference associated with bj-notbi 10716. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
Theorem	bj-notbid 10718	Deduction form of bj-notbi 10716. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
Theorem	bj-dcbi 10719	Equivalence property for DECID. TODO: solve conflict with dcbi 877; minimize dcbii 780 and dcbid 781 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))
Theorem	bj-d0clsepcl 10720	Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
⊢ DECID 𝜑
6.2.5.2  Inductive classes and the class of natural numbers (finite ordinals)
Syntax	wind 10721	Syntax for inductive classes.
wff Ind 𝐴
Definition	df-bj-ind 10722*	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 10723	A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
Theorem	bj-indeq 10724	Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
Theorem	bj-bdind 10725	Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
⊢ BOUNDED Ind 𝑥
Theorem	bj-indint 10726*	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 10727*	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 10728	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 10729	ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
⊢ Ind ω
Theorem	bj-omssind 10730	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 10731*	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 10732*	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 10733*	Two formulations of the axiom of infinity (see ax-infvn 10736 and bj-omex 10737) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))
6.2.5.3  The first three Peano postulates The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4335 and peano3 4337 already show this. In this section, we prove bj-peano2 10734 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.
Theorem	bj-peano2 10734	Constructive proof of peano2 4336. Temporary note: another possibility is to simply replace sucexg 4242 with bj-sucexg 10713 in the proof of peano2 4336. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
Theorem	peano5set 10735*	Version of peano5 4339 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))
6.2.6  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.
6.2.6.1  The set of natural numbers (finite ordinals) In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 10736) and deduce that the class ω of finite ordinals is a set (bj-omex 10737).
Axiom	ax-infvn 10736*	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 4329) from which one then proves, using full separation, that the wanted set exists (omex 4334). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))
Theorem	bj-omex 10737	Proof of omex 4334 from ax-infvn 10736. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
⊢ ω ∈ V
6.2.6.2  Peano's fifth postulate In this section, we give constructive proofs of two versions of Peano's fifth postulate.
Theorem	bdpeano5 10738*	Bounded version of peano5 4339. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ BOUNDED 𝐴    ⇒   ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
Theorem	speano5 10739*	Version of peano5 4339 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)
6.2.6.3  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 finite ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.
Theorem	findset 10740*	Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4340 for a nonconstructive proof of the general case. See bdfind 10741 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 10741*	Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4340 for a nonconstructive proof of the general case. See findset 10740 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 10742*	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 4341 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4341, finds2 4342, finds1 4343. (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 10743*	Version of bj-bdfindis 10742 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 10742 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 10744	Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 10742 for explanations. From this version, it is easy to prove the bounded version of findes 4344. (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 10745*	Constructive proof of a variant of nn0suc 4345. For a constructive proof of nn0suc 4345, see bj-nn0suc 10759. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
Theorem	bj-nntrans 10746	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 10747	A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Tr 𝐴)
Theorem	bj-nnelirr 10748	A natural number does not belong to itself. Version of elirr 4284 for natural numbers, which does not require ax-setind 4280. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)
Theorem	bj-nnen2lp 10749	A version of en2lp 4297 for natural numbers, which does not require ax-setind 4280. Note: using this theorem and bj-nnelirr 10748, one can remove dependency on ax-setind 4280 from nntri2 6096 and nndcel 6101; 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 10750	Remove from peano4 4338 dependency on ax-setind 4280. 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 10751	The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4346. 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 10752	The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Tr ω
Theorem	bj-nnord 10753	A natural number is an ordinal. Constructive proof of nnord 4352. Can also be proved from bj-nnelon 10754 if the latter is proved from bj-omssonALT 10758. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	bj-nnelon 10754	A natural number is an ordinal. Constructive proof of nnon 4350. Can also be proved from bj-omssonALT 10758. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	bj-omord 10755	The set ω is an ordinal. Constructive proof of ordom 4347. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Ord ω
Theorem	bj-omelon 10756	The set ω is an ordinal. Constructive proof of omelon 4349. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ ω ∈ On
Theorem	bj-omsson 10757	Constructive proof of omsson 4353. See also bj-omssonALT 10758. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
⊢ ω ⊆ On
Theorem	bj-omssonALT 10758	Alternate proof of bj-omsson 10757. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ⊆ On
Theorem	bj-nn0suc 10759*	Proof of (biconditional form of) nn0suc 4345 from the core axioms of CZF. See also bj-nn0sucALT 10773. 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))
6.2.7  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
6.2.7.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 10760*	Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑))
Theorem	setindf 10761*	Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)
Theorem	setindis 10762*	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 10763*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
Theorem	bdsetindis 10764*	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 10765*	Lemma for bj-inf2vn 10769. 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 10766*	Lemma for bj-inf2vnlem3 10767 and bj-inf2vnlem4 10768. 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 10767*	Lemma for bj-inf2vn 10769. (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 10768*	Lemma for bj-inf2vn2 10770. (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 10769*	A sufficient condition for ω to be a set. See bj-inf2vn2 10770 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 10770*	A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 10769. (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 10771*	Another axiom of infinity in a constructive setting (see ax-infvn 10736). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
Theorem	bj-omex2 10772	Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 10736 (see bj-2inf 10733 for the equivalence of the latter with bj-omex 10737). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	bj-nn0sucALT 10773*	Alternate proof of bj-nn0suc 10759, also constructive but from ax-inf2 10771, hence requiring ax-bdsetind 10763. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
6.2.7.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 10774*	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 10742 for a bounded version not requiring ax-setind 4280. See finds 4341 for a proof in IZF. From this version, it is easy to prove of finds 4341, finds2 4342, finds1 4343. (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 10775*	Version of bj-findis 10774 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10774 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 10776	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 10774 for explanations. From this version, it is easy to prove findes 4344. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
6.2.8  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 10777*	Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcoll2 10778*	Version of ax-strcoll 10777 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcollnft 10779*	Closed form of strcollnf 10780. Version of ax-strcoll 10777 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
Theorem	strcollnf 10780*	Version of ax-strcoll 10777 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
⊢ Ⅎ𝑏𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcollnfALT 10781*	Alternate proof of strcollnf 10780, not using strcollnft 10779. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑏𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))
6.2.9  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 10782*	Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.)
⊢ ∀𝑎∀𝑏∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
Theorem	sscoll2 10783*	Version of ax-sscoll 10782 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 ∀𝑦(𝑦 ∈ 𝑑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
6.2.10  Real numbers
Axiom	ax-ddkcomp 10784	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 the axiom ax-ddkcomp 10784 should be used in place of construction specific results. In particular, axcaucvg 7066 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)))
6.3  Mathbox for Jim Kingdon
Theorem	qdencn 10785*	The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 10088 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
⊢ 𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)}    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ 𝑄 (abs‘(𝑥 − 𝐴)) < 𝐵)
6.4  Mathbox for Mykola Mostovenko
Theorem	ax1hfs 10786	Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)
⊢ (𝜑 → (𝜑 ∧ 𝜑))
6.5  Mathbox for David A. Wheeler
6.5.1  Allsome quantifier These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like ∀𝑥𝜑 → 𝜓 do not imply that 𝜑 is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ∀!𝑥(𝜑 → 𝜓), and when restricted (applied to a class) we allow ∀!𝑥 ∈ 𝐴𝜑. The first symbol after the setvar variable must always be ∈ if it is the form applied to a class, and since ∈ cannot begin a wff, it is unambiguous. The → looks like it would be a problem because 𝜑 or 𝜓 might include implications, but any implication arrow → within any wff must be surrounded by parentheses, so only the implication arrow of ∀! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome.
Syntax	walsi 10787	Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff ∀!𝑥(𝜑 → 𝜓)
Syntax	walsc 10788	Extend wff definition to include "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.)
wff ∀!𝑥 ∈ 𝐴𝜑
Definition	df-alsi 10789	Define "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
Definition	df-alsc 10790	Define "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴 and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴))
Theorem	alsconv 10791	There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.)
⊢ (∀!𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀!𝑥 ∈ 𝐴𝜑)
Theorem	alsi1d 10792	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
Theorem	alsi2d 10793	Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (𝜑 → ∀!𝑥(𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	alsc1d 10794	Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	alsc2d 10795	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
⊢ (𝜑 → ∀!𝑥 ∈ 𝐴𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

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