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Theorem List for Intuitionistic Logic Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembdctp 11201 The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥, 𝑦, 𝑧}

Theorembdsnss 11202* Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED {𝑥} ⊆ 𝐴

Theorembdvsn 11203* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥 = {𝑦}

Theorembdop 11204 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED𝑥, 𝑦

Theorembdcuni 11205 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
BOUNDED 𝑥

Theorembdcint 11206 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥

Theorembdciun 11207* The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝑦 𝐴

Theorembdciin 11208* The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝑦 𝐴

Theorembdcsuc 11209 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED suc 𝑥

Theorembdeqsuc 11210* Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
BOUNDED 𝑥 = suc 𝑦

Theorembj-bdsucel 11211 Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.)
BOUNDED suc 𝑥𝑦

Theorembdcriota 11212* A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
BOUNDED 𝜑    &   ∃!𝑥𝑦 𝜑       BOUNDED (𝑥𝑦 𝜑)

6.2.7  CZF: Bounded separation

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

Axiomax-bdsep 11213* Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3932. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑎𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))

Theorembdsep1 11214* Version of ax-bdsep 11213 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))

Theorembdsep2 11215* Version of ax-bdsep 11213 with one DV condition removed and without initial universal quantifier. Use bdsep1 11214 when sufficient. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))

Theorembdsepnft 11216* Closed form of bdsepnf 11217. Version of ax-bdsep 11213 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 11214 when sufficient. (Contributed by BJ, 19-Oct-2019.)
BOUNDED 𝜑       (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))

Theorembdsepnf 11217* Version of ax-bdsep 11213 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 11218. Use bdsep1 11214 when sufficient. (Contributed by BJ, 5-Oct-2019.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))

TheorembdsepnfALT 11218* Alternate proof of bdsepnf 11217, not using bdsepnft 11216. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))

Theorembdzfauscl 11219* Closed form of the version of zfauscl 3934 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
BOUNDED 𝜑       (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))

Theorembdbm1.3ii 11220* Bounded version of bm1.3ii 3935. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)

Theorembj-axemptylem 11221* Lemma for bj-axempty 11222 and bj-axempty2 11223. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3940 instead. (New usage is discouraged.)
𝑥𝑦(𝑦𝑥 → ⊥)

Theorembj-axempty 11222* 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 3939. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3940 instead. (New usage is discouraged.)
𝑥𝑦𝑥

Theorembj-axempty2 11223* Axiom of the empty set from bounded separation, alternate version to bj-axempty 11222. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3940 instead. (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥

Theorembj-nalset 11224* nalset 3944 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥

Theorembj-vprc 11225 vprc 3946 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ V

Theorembj-nvel 11226 nvel 3947 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ 𝐴

Theorembj-vnex 11227 vnex 3945 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥 𝑥 = V

Theorembdinex1 11228 Bounded version of inex1 3948. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐴𝐵) ∈ V

Theorembdinex2 11229 Bounded version of inex2 3949. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐵𝐴) ∈ V

Theorembdinex1g 11230 Bounded version of inex1g 3950. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵       (𝐴𝑉 → (𝐴𝐵) ∈ V)

Theorembdssex 11231 Bounded version of ssex 3951. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theorembdssexi 11232 Bounded version of ssexi 3952. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theorembdssexg 11233 Bounded version of ssexg 3953. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theorembdssexd 11234 Bounded version of ssexd 3954. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)    &   BOUNDED 𝐴       (𝜑𝐴 ∈ V)

Theorembdrabexg 11235* Bounded version of rabexg 3957. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   BOUNDED 𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theorembj-inex 11236 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 11237 intexr 3961 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)

Theorembj-intnexr 11238 intnexr 3962 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 = V → ¬ 𝐴 ∈ V)

Theorembj-zfpair2 11239 Proof of zfpair2 4011 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V

Theorembj-prexg 11240 Proof of prexg 4012 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)

Theorembj-snexg 11241 snexg 3993 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)

Theorembj-snex 11242 snex 3994 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V

Theorembj-sels 11243* 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 11244* axun2 4236 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))

Theorembj-uniex2 11245* uniex2 4237 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥

Theorembj-uniex 11246 uniex 4238 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V

Theorembj-uniexg 11247 uniexg 4239 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)

Theorembj-unex 11248 unex 4240 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V

Theorembdunexb 11249 Bounded version of unexb 4241. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Theorembj-unexg 11250 unexg 4242 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theorembj-sucexg 11251 sucexg 4288 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)

Theorembj-sucex 11252 sucex 4289 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       suc 𝐴 ∈ V

6.2.7.1  Delta_0-classical logic

Axiomax-bj-d0cl 11253 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑

Theorembj-notbi 11254 Equivalence property for negation. TODO: minimize all theorems using notbid 625 and notbii 627. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Theorembj-notbii 11255 Inference associated with bj-notbi 11254. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)

Theorembj-notbid 11256 Deduction form of bj-notbi 11254. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))

Theorembj-dcbi 11257 Equivalence property for DECID. TODO: solve conflict with dcbi 880; minimize dcbii 783 and dcbid 784 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (DECID 𝜑DECID 𝜓))

Theorembj-d0clsepcl 11258 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID 𝜑

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

Syntaxwind 11259 Syntax for inductive classes.
wff Ind 𝐴

Definitiondf-bj-ind 11260* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))

Theorembj-indsuc 11261 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))

Theorembj-indeq 11262 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))

Theorembj-bdind 11263 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥

Theorembj-indint 11264* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}

Theorembj-indind 11265* 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 11266 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 11267 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω

Theorembj-omssind 11268 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 11269* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)

Theorembj-om 11270* 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 11271* Two formulations of the axiom of infinity (see ax-infvn 11274 and bj-omex 11275) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))

6.2.7.3  The first three Peano postulates

The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4382 and peano3 4384 already show this. In this section, we prove bj-peano2 11272 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

Theorembj-peano2 11272 Constructive proof of peano2 4383. Temporary note: another possibility is to simply replace sucexg 4288 with bj-sucexg 11251 in the proof of peano2 4383. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)

Theorempeano5set 11273* Version of peano5 4386 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))

6.2.8  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.

6.2.8.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 11274) and deduce that the class ω of finite ordinals is a set (bj-omex 11275).

Axiomax-infvn 11274* 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 4376) from which one then proves, using full separation, that the wanted set exists (omex 4381). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))

Theorembj-omex 11275 Proof of omex 4381 from ax-infvn 11274. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
ω ∈ V

6.2.8.2  Peano's fifth postulate

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

Theorembdpeano5 11276* Bounded version of peano5 4386. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)

Theoremspeano5 11277* Version of peano5 4386 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)

6.2.8.3  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 11278* Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4387 for a nonconstructive proof of the general case. See bdfind 11279 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 11279* Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4387 for a nonconstructive proof of the general case. See findset 11278 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 11280* 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 4388 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4388, finds2 4389, finds1 4390. (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 11281* Version of bj-bdfindis 11280 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 11280 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 11282 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 11280 for explanations. From this version, it is easy to prove the bounded version of findes 4391. (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 11283* Constructive proof of a variant of nn0suc 4392. For a constructive proof of nn0suc 4392, see bj-nn0suc 11297. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))

Theorembj-nntrans 11284 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))

Theorembj-nntrans2 11285 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Tr 𝐴)

Theorembj-nnelirr 11286 A natural number does not belong to itself. Version of elirr 4330 for natural numbers, which does not require ax-setind 4326. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → ¬ 𝐴𝐴)

Theorembj-nnen2lp 11287 A version of en2lp 4343 for natural numbers, which does not require ax-setind 4326.

Note: using this theorem and bj-nnelirr 11286, one can remove dependency on ax-setind 4326 from nntri2 6209 and nndcel 6215; 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 11288 Remove from peano4 4385 dependency on ax-setind 4326. 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 11289 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4393.

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 11290 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr ω

Theorembj-nnord 11291 A natural number is an ordinal. Constructive proof of nnord 4399. Can also be proved from bj-nnelon 11292 if the latter is proved from bj-omssonALT 11296. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)

Theorembj-nnelon 11292 A natural number is an ordinal. Constructive proof of nnon 4397. Can also be proved from bj-omssonALT 11296. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)

Theorembj-omord 11293 The set ω is an ordinal. Constructive proof of ordom 4394. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Ord ω

Theorembj-omelon 11294 The set ω is an ordinal. Constructive proof of omelon 4396. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
ω ∈ On

Theorembj-omsson 11295 Constructive proof of omsson 4400. See also bj-omssonALT 11296. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On

Theorembj-omssonALT 11296 Alternate proof of bj-omsson 11295. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On

Theorembj-nn0suc 11297* Proof of (biconditional form of) nn0suc 4392 from the core axioms of CZF. See also bj-nn0sucALT 11311. 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))

6.2.9  CZF: Set induction

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

6.2.9.1  Set induction

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

Theoremsetindft 11298* 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 11299* 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 11300* 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)

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