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Theorem List for Intuitionistic Logic Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-axempty2 10401* Axiom of the empty set from bounded separation, alternate version to bj-axempty 10400. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3911 instead. (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥

Theorembj-nalset 10402* nalset 3915 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥

Theorembj-vprc 10403 vprc 3916 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ V

Theorembj-nvel 10404 nvel 3917 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ 𝐴

Theorembj-vnex 10405 vnex 3918 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥 𝑥 = V

Theorembdinex1 10406 Bounded version of inex1 3919. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐴𝐵) ∈ V

Theorembdinex2 10407 Bounded version of inex2 3920. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐵𝐴) ∈ V

Theorembdinex1g 10408 Bounded version of inex1g 3921. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵       (𝐴𝑉 → (𝐴𝐵) ∈ V)

Theorembdssex 10409 Bounded version of ssex 3922. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theorembdssexi 10410 Bounded version of ssexi 3923. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theorembdssexg 10411 Bounded version of ssexg 3924. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theorembdssexd 10412 Bounded version of ssexd 3925. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)    &   BOUNDED 𝐴       (𝜑𝐴 ∈ V)

Theorembdrabexg 10413* Bounded version of rabexg 3928. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   BOUNDED 𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theorembj-inex 10414 The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theorembj-intexr 10415 intexr 3932 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)

Theorembj-intnexr 10416 intnexr 3933 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 = V → ¬ 𝐴 ∈ V)

Theorembj-zfpair2 10417 Proof of zfpair2 3973 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V

Theorembj-prexg 10418 Proof of prexg 3975 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)

Theorembj-snexg 10419 snexg 3964 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)

Theorembj-snex 10420 snex 3965 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V

Theorembj-sels 10421* If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
(𝐴𝑉 → ∃𝑥 𝐴𝑥)

Theorembj-axun2 10422* axun2 4200 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))

Theorembj-uniex2 10423* uniex2 4201 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥

Theorembj-uniex 10424 uniex 4202 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V

Theorembj-uniexg 10425 uniexg 4203 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)

Theorembj-unex 10426 unex 4204 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V

Theorembdunexb 10427 Bounded version of unexb 4205. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Theorembj-unexg 10428 unexg 4206 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theorembj-sucexg 10429 sucexg 4252 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)

Theorembj-sucex 10430 sucex 4253 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       suc 𝐴 ∈ V

6.3.6.1  Delta_0-classical logic

Axiomax-bj-d0cl 10431 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑

Theorembj-notbi 10432 Equivalence property for negation. TODO: minimize all theorems using notbid 602 and notbii 604. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Theorembj-notbii 10433 Inference associated with bj-notbi 10432. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)

Theorembj-notbid 10434 Deduction form of bj-notbi 10432. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))

Theorembj-dcbi 10435 Equivalence property for DECID. TODO: solve conflict with dcbi 855; minimize dcbii 758 and dcbid 759 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (DECID 𝜑DECID 𝜓))

Theorembj-d0clsepcl 10436 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID 𝜑

6.3.6.2  Inductive classes and the class of natural numbers (finite ordinals)

Syntaxwind 10437 Syntax for inductive classes.
wff Ind 𝐴

Definitiondf-bj-ind 10438* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))

Theorembj-indsuc 10439 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))

Theorembj-indeq 10440 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))

Theorembj-bdind 10441 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥

Theorembj-indint 10442* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}

Theorembj-indind 10443* If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))

Theorembj-dfom 10444 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}

Theorembj-omind 10445 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω

Theorembj-omssind 10446 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.)
(𝐴𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))

Theorembj-ssom 10447* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)

Theorembj-om 10448* 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.)
(𝐴𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥𝐴𝑥))))

Theorembj-2inf 10449* Two formulations of the axiom of infinity (see ax-infvn 10453 and bj-omex 10454) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))

6.3.6.3  The first three Peano postulates

The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4345 and peano3 4347 already show this. In this section, we prove bj-peano2 10450 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

Theorembj-peano2 10450 Constructive proof of peano2 4346. Temporary note: another possibility is to simply replace sucexg 4252 with bj-sucexg 10429 in the proof of peano2 4346. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)

Theorempeano5set 10451* Version of peano5 4349 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 (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴))

Theorempeano5setOLD 10452* Obsolete version of peano5set 10451 as of 26-Oct-2020. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴))

6.3.7  Axiom of 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.

6.3.7.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 10453) and deduce that the class ω of finite ordinals is a set (bj-omex 10454).

Axiomax-infvn 10453* 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 4339) from which one then proves, using full separation, that the wanted set exists (omex 4344). "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.)
𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))

Theorembj-omex 10454 Proof of omex 4344 from ax-infvn 10453. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
ω ∈ V

6.3.7.2  Peano's fifth postulate

In this section, we give constructive proofs of two versions of Peano's fifth postulate.

Theorembdpeano5 10455* Bounded version of peano5 4349. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)

Theoremspeano5 10456* Version of peano5 4349 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.)
((𝐴𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)

6.3.7.3  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.

Theoremfindset 10457* Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4350 for a nonconstructive proof of the general case. See bdfind 10458 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω))

Theorembdfind 10458* Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4350 for a nonconstructive proof of the general case. See findset 10457 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 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω)

Theorembj-bdfindis 10459* 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 4351 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4351, finds2 4352, finds1 4353. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝜓    &   𝑥𝜒    &   𝑥𝜃    &   (𝑥 = ∅ → (𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜃𝜑))       ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)

Theorembj-bdfindisg 10460* Version of bj-bdfindis 10459 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 10459 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝜓    &   𝑥𝜒    &   𝑥𝜃    &   (𝑥 = ∅ → (𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜃𝜑))    &   𝑥𝐴    &   𝑥𝜏    &   (𝑥 = 𝐴 → (𝜑𝜏))       ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (𝐴 ∈ ω → 𝜏))

Theorembj-bdfindes 10461 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 10459 for explanations. From this version, it is easy to prove the bounded version of findes 4354. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑       (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

Theorembj-nn0suc0 10462* Constructive proof of a variant of nn0suc 4355. For a constructive proof of nn0suc 4355, see bj-nn0suc 10476. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))

Theorembj-nntrans 10463 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))

Theorembj-nntrans2 10464 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Tr 𝐴)

Theorembj-nnelirr 10465 A natural number does not belong to itself. Version of elirr 4294 for natural numbers, which does not require ax-setind 4290. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → ¬ 𝐴𝐴)

Theorembj-nnen2lp 10466 A version of en2lp 4306 for natural numbers, which does not require ax-setind 4290.

Note: using this theorem and bj-nnelirr 10465, one can remove dependency on ax-setind 4290 from nntri2 6104 and nndcel 6109; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴𝐵𝐵𝐴))

Theorembj-peano4 10467 Remove from peano4 4348 dependency on ax-setind 4290. 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 𝐵𝐴 = 𝐵))

Theorembj-omtrans 10468 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4356.

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 𝑥𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

(𝐴 ∈ ω → 𝐴 ⊆ ω)

Theorembj-omtrans2 10469 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr ω

Theorembj-nnord 10470 A natural number is an ordinal. Constructive proof of nnord 4362. Can also be proved from bj-nnelon 10471 if the latter is proved from bj-omssonALT 10475. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)

Theorembj-nnelon 10471 A natural number is an ordinal. Constructive proof of nnon 4360. Can also be proved from bj-omssonALT 10475. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)

Theorembj-omord 10472 The set ω is an ordinal. Constructive proof of ordom 4357. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Ord ω

Theorembj-omelon 10473 The set ω is an ordinal. Constructive proof of omelon 4359. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
ω ∈ On

Theorembj-omsson 10474 Constructive proof of omsson 4363. See also bj-omssonALT 10475. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On

Theorembj-omssonALT 10475 Alternate proof of bj-omsson 10474. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On

Theorembj-nn0suc 10476* Proof of (biconditional form of) nn0suc 4355 from the core axioms of CZF. See also bj-nn0sucALT 10490. 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))

6.3.8  Set induction

In this section, we add the axiom of set induction to the core axioms of CZF.

6.3.8.1  Set induction

In this section, we prove some variants of the axiom of set induction.

Theoremsetindft 10477* Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))

Theoremsetindf 10478* Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
𝑦𝜑       (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)

Theoremsetindis 10479* Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
𝑥𝜓    &   𝑥𝜒    &   𝑦𝜑    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜒𝜑))       (∀𝑦(∀𝑧𝑦 𝜓𝜒) → ∀𝑥𝜑)

Axiomax-bdsetind 10480* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
BOUNDED 𝜑       (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)

Theorembdsetindis 10481* Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝜓    &   𝑥𝜒    &   𝑦𝜑    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜒𝜑))       (∀𝑦(∀𝑧𝑦 𝜓𝜒) → ∀𝑥𝜑)

Theorembj-inf2vnlem1 10482* Lemma for bj-inf2vn 10486. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)

Theorembj-inf2vnlem2 10483* Lemma for bj-inf2vnlem3 10484 and bj-inf2vnlem4 10485. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡𝑢 (𝑡𝐴𝑡𝑍) → (𝑢𝐴𝑢𝑍))))

Theorembj-inf2vnlem3 10484* Lemma for bj-inf2vn 10486. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝑍       (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))

Theorembj-inf2vnlem4 10485* Lemma for bj-inf2vn2 10487. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))

Theorembj-inf2vn 10486* A sufficient condition for ω to be a set. See bj-inf2vn2 10487 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       (𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Theorembj-inf2vn2 10487* A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 10486. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Axiomax-inf2 10488* Another axiom of infinity in a constructive setting (see ax-infvn 10453). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))

Theorembj-omex2 10489 Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 10453 (see bj-2inf 10449 for the equivalence of the latter with bj-omex 10454). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ∈ V

Theorembj-nn0sucALT 10490* Alternate proof of bj-nn0suc 10476, also constructive but from ax-inf2 10488, hence requiring ax-bdsetind 10480. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

6.3.8.2  Full induction

In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.

Theorembj-findis 10491* 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 10459 for a bounded version not requiring ax-setind 4290. See finds 4351 for a proof in IZF. From this version, it is easy to prove of finds 4351, finds2 4352, finds1 4353. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝑥𝜒    &   𝑥𝜃    &   (𝑥 = ∅ → (𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜃𝜑))       ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)

Theorembj-findisg 10492* Version of bj-findis 10491 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10491 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   𝑥𝜒    &   𝑥𝜃    &   (𝑥 = ∅ → (𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜃𝜑))    &   𝑥𝐴    &   𝑥𝜏    &   (𝑥 = 𝐴 → (𝜑𝜏))       ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (𝐴 ∈ ω → 𝜏))

Theorembj-findes 10493 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 10491 for explanations. From this version, it is easy to prove findes 4354. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
(([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

6.3.9  Strong collection

In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.

Axiomax-strcoll 10494* 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(∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

Theoremstrcoll2 10495* Version of ax-strcoll 10494 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
(∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

Theoremstrcollnft 10496* Closed form of strcollnf 10497. Version of ax-strcoll 10494 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.)
(∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))

Theoremstrcollnf 10497* Version of ax-strcoll 10494 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.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

TheoremstrcollnfALT 10498* Alternate proof of strcollnf 10497, not using strcollnft 10496. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

6.3.10  Subset collection

In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.

Axiomax-sscoll 10499* 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.)
𝑎𝑏𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))

Theoremsscoll2 10500* Version of ax-sscoll 10499 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))

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