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Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdss 11401 The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝑥𝐴
 
Theorembdcnul 11402 The empty class is bounded. See also bdcnulALT 11403. (Contributed by BJ, 3-Oct-2019.)
BOUNDED
 
TheorembdcnulALT 11403 Alternate proof of bdcnul 11402. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 11381, or use the corresponding characterizations of its elements followed by bdelir 11384. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
BOUNDED
 
Theorembdeq0 11404 Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED 𝑥 = ∅
 
Theorembj-bd0el 11405 Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
BOUNDED ∅ ∈ 𝑥
 
Theorembdcpw 11406 The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
BOUNDED 𝐴       BOUNDED 𝒫 𝐴
 
Theorembdcsn 11407 The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥}
 
Theorembdcpr 11408 The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥, 𝑦}
 
Theorembdctp 11409 The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
BOUNDED {𝑥, 𝑦, 𝑧}
 
Theorembdsnss 11410* 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 11411* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥 = {𝑦}
 
Theorembdop 11412 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
BOUNDED𝑥, 𝑦
 
Theorembdcuni 11413 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
BOUNDED 𝑥
 
Theorembdcint 11414 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED 𝑥
 
Theorembdciun 11415* 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 11416* 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 11417 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
BOUNDED suc 𝑥
 
Theorembdeqsuc 11418* 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 11419 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 11420* 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 11421* 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 3949. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑎𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdsep1 11422* Version of ax-bdsep 11421 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdsep2 11423* Version of ax-bdsep 11421 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 11422 when sufficient. (Contributed by BJ, 5-Oct-2019.)
BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdsepnft 11424* Closed form of bdsepnf 11425. Version of ax-bdsep 11421 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 11422 when sufficient. (Contributed by BJ, 19-Oct-2019.)
BOUNDED 𝜑       (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
 
Theorembdsepnf 11425* Version of ax-bdsep 11421 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 11426. Use bdsep1 11422 when sufficient. (Contributed by BJ, 5-Oct-2019.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
TheorembdsepnfALT 11426* Alternate proof of bdsepnf 11425, not using bdsepnft 11424. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑    &   BOUNDED 𝜑       𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 
Theorembdzfauscl 11427* Closed form of the version of zfauscl 3951 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
BOUNDED 𝜑       (𝐴𝑉 → ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
 
Theorembdbm1.3ii 11428* Bounded version of bm1.3ii 3952. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)
 
Theorembj-axemptylem 11429* Lemma for bj-axempty 11430 and bj-axempty2 11431. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3957 instead. (New usage is discouraged.)
𝑥𝑦(𝑦𝑥 → ⊥)
 
Theorembj-axempty 11430* 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 3956. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3957 instead. (New usage is discouraged.)
𝑥𝑦𝑥
 
Theorembj-axempty2 11431* Axiom of the empty set from bounded separation, alternate version to bj-axempty 11430. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3957 instead. (New usage is discouraged.)
𝑥𝑦 ¬ 𝑦𝑥
 
Theorembj-nalset 11432* nalset 3961 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥𝑦 𝑦𝑥
 
Theorembj-vprc 11433 vprc 3963 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ V
 
Theorembj-nvel 11434 nvel 3964 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ V ∈ 𝐴
 
Theorembj-vnex 11435 vnex 3962 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
¬ ∃𝑥 𝑥 = V
 
Theorembdinex1 11436 Bounded version of inex1 3965. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdinex2 11437 Bounded version of inex2 3966. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵    &   𝐴 ∈ V       (𝐵𝐴) ∈ V
 
Theorembdinex1g 11438 Bounded version of inex1g 3967. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐵       (𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theorembdssex 11439 Bounded version of ssex 3968. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)
 
Theorembdssexi 11440 Bounded version of ssexi 3969. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V
 
Theorembdssexg 11441 Bounded version of ssexg 3970. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorembdssexd 11442 Bounded version of ssexd 3971. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)    &   BOUNDED 𝐴       (𝜑𝐴 ∈ V)
 
Theorembdrabexg 11443* Bounded version of rabexg 3974. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   BOUNDED 𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theorembj-inex 11444 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 11445 intexr 3978 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)
 
Theorembj-intnexr 11446 intnexr 3979 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 = V → ¬ 𝐴 ∈ V)
 
Theorembj-zfpair2 11447 Proof of zfpair2 4028 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V
 
Theorembj-prexg 11448 Proof of prexg 4029 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
 
Theorembj-snexg 11449 snexg 4010 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)
 
Theorembj-snex 11450 snex 4011 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V
 
Theorembj-sels 11451* 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 11452* axun2 4253 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theorembj-uniex2 11453* uniex2 4254 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥
 
Theorembj-uniex 11454 uniex 4255 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V
 
Theorembj-uniexg 11455 uniexg 4256 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)
 
Theorembj-unex 11456 unex 4257 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdunexb 11457 Bounded version of unexb 4258. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
Theorembj-unexg 11458 unexg 4259 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theorembj-sucexg 11459 sucexg 4305 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theorembj-sucex 11460 sucex 4306 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 11461 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑
 
Theorembj-notbi 11462 Equivalence property for negation. TODO: minimize all theorems using notbid 627 and notbii 629. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
 
Theorembj-notbii 11463 Inference associated with bj-notbi 11462. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)
 
Theorembj-notbid 11464 Deduction form of bj-notbi 11462. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
 
Theorembj-dcbi 11465 Equivalence property for DECID. TODO: solve conflict with dcbi 882; minimize dcbii 785 and dcbid 786 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → (DECID 𝜑DECID 𝜓))
 
Theorembj-d0clsepcl 11466 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 11467 Syntax for inductive classes.
wff Ind 𝐴
 
Definitiondf-bj-ind 11468* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
 
Theorembj-indsuc 11469 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
 
Theorembj-indeq 11470 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
 
Theorembj-bdind 11471 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥
 
Theorembj-indint 11472* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}
 
Theorembj-indind 11473* 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 11474 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 11475 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω
 
Theorembj-omssind 11476 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 11477* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
 
Theorembj-om 11478* 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 11479* Two formulations of the axiom of infinity (see ax-infvn 11482 and bj-omex 11483) . (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 4399 and peano3 4401 already show this. In this section, we prove bj-peano2 11480 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

 
Theorembj-peano2 11480 Constructive proof of peano2 4400. Temporary note: another possibility is to simply replace sucexg 4305 with bj-sucexg 11459 in the proof of peano2 4400. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)
 
Theorempeano5set 11481* Version of peano5 4403 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 11482) and deduce that the class ω of finite ordinals is a set (bj-omex 11483).

 
Axiomax-infvn 11482* 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 4393) from which one then proves, using full separation, that the wanted set exists (omex 4398). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))
 
Theorembj-omex 11483 Proof of omex 4398 from ax-infvn 11482. (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 11484* Bounded version of peano5 4403. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
 
Theoremspeano5 11485* Version of peano5 4403 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 11486* Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4404 for a nonconstructive proof of the general case. See bdfind 11487 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 11487* Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4404 for a nonconstructive proof of the general case. See findset 11486 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 11488* 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 4405 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4405, finds2 4406, finds1 4407. (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 11489* Version of bj-bdfindis 11488 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 11488 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 11490 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 11488 for explanations. From this version, it is easy to prove the bounded version of findes 4408. (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 11491* Constructive proof of a variant of nn0suc 4409. For a constructive proof of nn0suc 4409, see bj-nn0suc 11505. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
 
Theorembj-nntrans 11492 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
 
Theorembj-nntrans2 11493 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Tr 𝐴)
 
Theorembj-nnelirr 11494 A natural number does not belong to itself. Version of elirr 4347 for natural numbers, which does not require ax-setind 4343. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → ¬ 𝐴𝐴)
 
Theorembj-nnen2lp 11495 A version of en2lp 4360 for natural numbers, which does not require ax-setind 4343.

Note: using this theorem and bj-nnelirr 11494, one can remove dependency on ax-setind 4343 from nntri2 6237 and nndcel 6243; 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 11496 Remove from peano4 4402 dependency on ax-setind 4343. 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 11497 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4410.

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 11498 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr ω
 
Theorembj-nnord 11499 A natural number is an ordinal. Constructive proof of nnord 4416. Can also be proved from bj-nnelon 11500 if the latter is proved from bj-omssonALT 11504. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)
 
Theorembj-nnelon 11500 A natural number is an ordinal. Constructive proof of nnon 4414. Can also be proved from bj-omssonALT 11504. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)
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