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Theorem List for Intuitionistic Logic Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheorembdsepnfALT 13101* Alternate proof of bdsepnf 13100, not using bdsepnft 13099. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdzfauscl 13102* Closed form of the version of zfauscl 4048 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
BOUNDED 𝜑       (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
 
Theorembdbm1.3ii 13103* Bounded version of bm1.3ii 4049. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)
 
Theorembj-axemptylem 13104* Lemma for bj-axempty 13105 and bj-axempty2 13106. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.)
𝑥𝑦(𝑦𝑥 → ⊥)
 
Theorembj-axempty 13105* 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 4053. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.)
𝑥𝑦𝑥
 
Theorembj-axempty2 13106* Axiom of the empty set from bounded separation, alternate version to bj-axempty 13105. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥
 
Theorembj-nalset 13107* nalset 4058 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theorembj-vprc 13108 vprc 4060 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ V
 
Theorembj-nvel 13109 nvel 4061 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ 𝐴
 
Theorembj-vnex 13110 vnex 4059 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥 𝑥 = V
 
Theorembdinex1 13111 Bounded version of inex1 4062. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdinex2 13112 Bounded version of inex2 4063. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐵𝐴) ∈ V
 
Theorembdinex1g 13113 Bounded version of inex1g 4064. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵       (𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theorembdssex 13114 Bounded version of ssex 4065. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)
 
Theorembdssexi 13115 Bounded version of ssexi 4066. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V
 
Theorembdssexg 13116 Bounded version of ssexg 4067. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorembdssexd 13117 Bounded version of ssexd 4068. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)    &   BOUNDED 𝐴       (𝜑𝐴 ∈ V)
 
Theorembdrabexg 13118* Bounded version of rabexg 4071. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   BOUNDED 𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theorembj-inex 13119 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 13120 intexr 4075 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)
 
Theorembj-intnexr 13121 intnexr 4076 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 = V → ¬ 𝐴 ∈ V)
 
Theorembj-zfpair2 13122 Proof of zfpair2 4132 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V
 
Theorembj-prexg 13123 Proof of prexg 4133 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
 
Theorembj-snexg 13124 snexg 4108 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)
 
Theorembj-snex 13125 snex 4109 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V
 
Theorembj-sels 13126* 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 13127* axun2 4357 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theorembj-uniex2 13128* uniex2 4358 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥
 
Theorembj-uniex 13129 uniex 4359 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V
 
Theorembj-uniexg 13130 uniexg 4361 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)
 
Theorembj-unex 13131 unex 4362 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdunexb 13132 Bounded version of unexb 4363. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
Theorembj-unexg 13133 unexg 4364 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theorembj-sucexg 13134 sucexg 4414 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theorembj-sucex 13135 sucex 4415 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
11.2.8.1  Delta_0-classical logic
 
Axiomax-bj-d0cl 13136 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑
 
Theorembj-d0clsepcl 13137 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID 𝜑
 
11.2.8.2  Inductive classes and the class of natural numbers (finite ordinals)
 
Syntaxwind 13138 Syntax for inductive classes.
wff Ind 𝐴
 
Definitiondf-bj-ind 13139* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
 
Theorembj-indsuc 13140 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
 
Theorembj-indeq 13141 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
 
Theorembj-bdind 13142 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥
 
Theorembj-indint 13143* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}
 
Theorembj-indind 13144* 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 13145 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 13146 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω
 
Theorembj-omssind 13147 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 13148* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
 
Theorembj-om 13149* 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 13150* Two formulations of the axiom of infinity (see ax-infvn 13153 and bj-omex 13154) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
 
11.2.8.3  The first three Peano postulates

The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4508 and peano3 4510 already show this. In this section, we prove bj-peano2 13151 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

 
Theorembj-peano2 13151 Constructive proof of peano2 4509. Temporary note: another possibility is to simply replace sucexg 4414 with bj-sucexg 13134 in the proof of peano2 4509. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)
 
Theorempeano5set 13152* Version of peano5 4512 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))
 
11.2.9  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.

 
11.2.9.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13153) and deduce that the class ω of finite ordinals is a set (bj-omex 13154).

 
Axiomax-infvn 13153* 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 4502) from which one then proves, using full separation, that the wanted set exists (omex 4507). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))
 
Theorembj-omex 13154 Proof of omex 4507 from ax-infvn 13153. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
ω ∈ V
 
11.2.9.2  Peano's fifth postulate

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

 
Theorembdpeano5 13155* Bounded version of peano5 4512. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
 
Theoremspeano5 13156* Version of peano5 4512 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)
 
11.2.9.3  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 13157* Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4513 for a nonconstructive proof of the general case. See bdfind 13158 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 13158* Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4513 for a nonconstructive proof of the general case. See findset 13157 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 13159* 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 4514 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4514, finds2 4515, finds1 4516. (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 13160* Version of bj-bdfindis 13159 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13159 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 13161 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13159 for explanations. From this version, it is easy to prove the bounded version of findes 4517. (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 13162* Constructive proof of a variant of nn0suc 4518. For a constructive proof of nn0suc 4518, see bj-nn0suc 13176. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
 
Theorembj-nntrans 13163 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
 
Theorembj-nntrans2 13164 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Tr 𝐴)
 
Theorembj-nnelirr 13165 A natural number does not belong to itself. Version of elirr 4456 for natural numbers, which does not require ax-setind 4452. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → ¬ 𝐴𝐴)
 
Theorembj-nnen2lp 13166 A version of en2lp 4469 for natural numbers, which does not require ax-setind 4452.

Note: using this theorem and bj-nnelirr 13165, one can remove dependency on ax-setind 4452 from nntri2 6390 and nndcel 6396; 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 13167 Remove from peano4 4511 dependency on ax-setind 4452. 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 13168 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4519.

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 13169 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr ω
 
Theorembj-nnord 13170 A natural number is an ordinal. Constructive proof of nnord 4525. Can also be proved from bj-nnelon 13171 if the latter is proved from bj-omssonALT 13175. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)
 
Theorembj-nnelon 13171 A natural number is an ordinal. Constructive proof of nnon 4523. Can also be proved from bj-omssonALT 13175. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)
 
Theorembj-omord 13172 The set ω is an ordinal. Constructive proof of ordom 4520. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Ord ω
 
Theorembj-omelon 13173 The set ω is an ordinal. Constructive proof of omelon 4522. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
ω ∈ On
 
Theorembj-omsson 13174 Constructive proof of omsson 4526. See also bj-omssonALT 13175. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On
 
Theorembj-omssonALT 13175 Alternate proof of bj-omsson 13174. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On
 
Theorembj-nn0suc 13176* Proof of (biconditional form of) nn0suc 4518 from the core axioms of CZF. See also bj-nn0sucALT 13190. 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))
 
11.2.10  CZF: Set induction

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

 
11.2.10.1  Set induction

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

 
Theoremsetindft 13177* Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))
 
Theoremsetindf 13178* Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
𝑦𝜑       (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
 
Theoremsetindis 13179* 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 13180* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
BOUNDED 𝜑       (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)
 
Theorembdsetindis 13181* 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 13182* Lemma for bj-inf2vn 13186. 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 13183* Lemma for bj-inf2vnlem3 13184 and bj-inf2vnlem4 13185. 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 13184* Lemma for bj-inf2vn 13186. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝑍       (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
 
Theorembj-inf2vnlem4 13185* Lemma for bj-inf2vn2 13187. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
 
Theorembj-inf2vn 13186* A sufficient condition for ω to be a set. See bj-inf2vn2 13187 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 13187* A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13186. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
 
Axiomax-inf2 13188* Another axiom of infinity in a constructive setting (see ax-infvn 13153). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))
 
Theorembj-omex2 13189 Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13153 (see bj-2inf 13150 for the equivalence of the latter with bj-omex 13154). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ∈ V
 
Theorembj-nn0sucALT 13190* Alternate proof of bj-nn0suc 13176, also constructive but from ax-inf2 13188, hence requiring ax-bdsetind 13180. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
 
11.2.10.2  Full induction

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

 
Theorembj-findis 13191* 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 13159 for a bounded version not requiring ax-setind 4452. See finds 4514 for a proof in IZF. From this version, it is easy to prove of finds 4514, finds2 4515, finds1 4516. (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 13192* Version of bj-findis 13191 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13191 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 13193 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13191 for explanations. From this version, it is easy to prove findes 4517. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
(([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
 
11.2.11  CZF: Strong collection

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

 
Axiomax-strcoll 13194* 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 13195* Version of ax-strcoll 13194 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
(∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
 
Theoremstrcollnft 13196* Closed form of strcollnf 13197. Version of ax-strcoll 13194 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
(∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
 
Theoremstrcollnf 13197* Version of ax-strcoll 13194 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
 
TheoremstrcollnfALT 13198* Alternate proof of strcollnf 13197, not using strcollnft 13196. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
 
11.2.12  CZF: Subset collection

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

 
Axiomax-sscoll 13199* 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 13200* Version of ax-sscoll 13199 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))
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