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Theorem List for Intuitionistic Logic Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdssexg 15401 Bounded version of ssexg 4168. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorembdssexd 15402 Bounded version of ssexd 4169. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)    &   BOUNDED 𝐴       (𝜑𝐴 ∈ V)
 
Theorembdrabexg 15403* Bounded version of rabexg 4172. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝜑    &   BOUNDED 𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theorembj-inex 15404 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 15405 intexr 4179 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)
 
Theorembj-intnexr 15406 intnexr 4180 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
( 𝐴 = V → ¬ 𝐴 ∈ V)
 
Theorembj-zfpair2 15407 Proof of zfpair2 4239 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V
 
Theorembj-prexg 15408 Proof of prexg 4240 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
 
Theorembj-snexg 15409 snexg 4213 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)
 
Theorembj-snex 15410 snex 4214 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V
 
Theorembj-sels 15411* 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 15412* axun2 4466 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theorembj-uniex2 15413* uniex2 4467 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥
 
Theorembj-uniex 15414 uniex 4468 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V
 
Theorembj-uniexg 15415 uniexg 4470 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)
 
Theorembj-unex 15416 unex 4472 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theorembdunexb 15417 Bounded version of unexb 4473. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
Theorembj-unexg 15418 unexg 4474 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theorembj-sucexg 15419 sucexg 4530 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theorembj-sucex 15420 sucex 4531 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
13.2.9.1  Delta_0-classical logic
 
Axiomax-bj-d0cl 15421 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑
 
Theorembj-d0clsepcl 15422 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID 𝜑
 
13.2.9.2  Inductive classes and the class of natural number ordinals
 
Syntaxwind 15423 Syntax for inductive classes.
wff Ind 𝐴
 
Definitiondf-bj-ind 15424* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
 
Theorembj-indsuc 15425 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
 
Theorembj-indeq 15426 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
 
Theorembj-bdind 15427 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥
 
Theorembj-indint 15428* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}
 
Theorembj-indind 15429* 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 15430 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 15431 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω
 
Theorembj-omssind 15432 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 15433* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)
 
Theorembj-om 15434* 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 15435* Two formulations of the axiom of infinity (see ax-infvn 15438 and bj-omex 15439) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))
 
13.2.9.3  The first three Peano postulates

The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4626 and peano3 4628 already show this. In this section, we prove bj-peano2 15436 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

 
Theorembj-peano2 15436 Constructive proof of peano2 4627. Temporary note: another possibility is to simply replace sucexg 4530 with bj-sucexg 15419 in the proof of peano2 4627. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)
 
Theorempeano5set 15437* Version of peano5 4630 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))
 
13.2.10  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.

 
13.2.10.1  The set of natural number ordinals

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 15438) and deduce that the class ω of natural number ordinals is a set (bj-omex 15439).

 
Axiomax-infvn 15438* 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 4620) from which one then proves, using full separation, that the wanted set exists (omex 4625). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))
 
Theorembj-omex 15439 Proof of omex 4625 from ax-infvn 15438. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
ω ∈ V
 
13.2.10.2  Peano's fifth postulate

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

 
Theorembdpeano5 15440* Bounded version of peano5 4630. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)
 
Theoremspeano5 15441* Version of peano5 4630 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)
 
13.2.10.3  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 natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers.

 
Theoremfindset 15442* Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4631 for a nonconstructive proof of the general case. See bdfind 15443 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 15443* Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4631 for a nonconstructive proof of the general case. See findset 15442 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 15444* 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 4632 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4632, finds2 4633, finds1 4634. (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 15445* Version of bj-bdfindis 15444 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 15444 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 15446 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 15444 for explanations. From this version, it is easy to prove the bounded version of findes 4635. (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 15447* Constructive proof of a variant of nn0suc 4636. For a constructive proof of nn0suc 4636, see bj-nn0suc 15461. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
 
Theorembj-nntrans 15448 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))
 
Theorembj-nntrans2 15449 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Tr 𝐴)
 
Theorembj-nnelirr 15450 A natural number does not belong to itself. Version of elirr 4573 for natural numbers, which does not require ax-setind 4569. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → ¬ 𝐴𝐴)
 
Theorembj-nnen2lp 15451 A version of en2lp 4586 for natural numbers, which does not require ax-setind 4569.

Note: using this theorem and bj-nnelirr 15450, one can remove dependency on ax-setind 4569 from nntri2 6547 and nndcel 6553; 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 15452 Remove from peano4 4629 dependency on ax-setind 4569. 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 15453 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4638.

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 15454 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr ω
 
Theorembj-nnord 15455 A natural number is an ordinal class. Constructive proof of nnord 4644. Can also be proved from bj-nnelon 15456 if the latter is proved from bj-omssonALT 15460. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)
 
Theorembj-nnelon 15456 A natural number is an ordinal. Constructive proof of nnon 4642. Can also be proved from bj-omssonALT 15460. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)
 
Theorembj-omord 15457 The set ω is an ordinal class. Constructive proof of ordom 4639. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Ord ω
 
Theorembj-omelon 15458 The set ω is an ordinal. Constructive proof of omelon 4641. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
ω ∈ On
 
Theorembj-omsson 15459 Constructive proof of omsson 4645. See also bj-omssonALT 15460. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On
 
Theorembj-omssonALT 15460 Alternate proof of bj-omsson 15459. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On
 
Theorembj-nn0suc 15461* Proof of (biconditional form of) nn0suc 4636 from the core axioms of CZF. See also bj-nn0sucALT 15475. 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))
 
13.2.11  CZF: Set induction

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

 
13.2.11.1  Set induction

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

 
Theoremsetindft 15462* Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑))
 
Theoremsetindf 15463* Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)
𝑦𝜑       (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
 
Theoremsetindis 15464* 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 15465* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
BOUNDED 𝜑       (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)
 
Theorembdsetindis 15466* 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 15467* Lemma for bj-inf2vn 15471. 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 15468* Lemma for bj-inf2vnlem3 15469 and bj-inf2vnlem4 15470. 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 15469* Lemma for bj-inf2vn 15471. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝑍       (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
 
Theorembj-inf2vnlem4 15470* Lemma for bj-inf2vn2 15472. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))
 
Theorembj-inf2vn 15471* A sufficient condition for ω to be a set. See bj-inf2vn2 15472 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 15472* A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 15471. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
 
Axiomax-inf2 15473* Another axiom of infinity in a constructive setting (see ax-infvn 15438). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))
 
Theorembj-omex2 15474 Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 15438 (see bj-2inf 15435 for the equivalence of the latter with bj-omex 15439). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ∈ V
 
Theorembj-nn0sucALT 15475* Alternate proof of bj-nn0suc 15461, also constructive but from ax-inf2 15473, hence requiring ax-bdsetind 15465. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
 
13.2.11.2  Full induction

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

 
Theorembj-findis 15476* 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 15444 for a bounded version not requiring ax-setind 4569. See finds 4632 for a proof in IZF. From this version, it is easy to prove of finds 4632, finds2 4633, finds1 4634. (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 15477* Version of bj-findis 15476 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 15476 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 15478 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15476 for explanations. From this version, it is easy to prove findes 4635. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
(([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
 
13.2.12  CZF: Strong collection

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

 
Axiomax-strcoll 15479* Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function on 𝑎, or equivalently a collection of nonempty classes indexed by 𝑎, and the axiom asserts the existence of a set 𝑏 which "collects" at least one element in the image of each 𝑥𝑎 and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4144. (Contributed by BJ, 5-Oct-2019.)
𝑎(∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 
Theoremstrcoll2 15480* Version of ax-strcoll 15479 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
(∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 
Theoremstrcollnft 15481* Closed form of strcollnf 15482. (Contributed by BJ, 21-Oct-2019.)
(∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
 
Theoremstrcollnf 15482* Version of ax-strcoll 15479 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 15480 with the disjoint variable condition on 𝑏, 𝜑 replaced with a nonfreeness hypothesis.

This proof aims to demonstrate a standard technique, but strcoll2 15480 will generally suffice: since the theorem asserts the existence of a set 𝑏, supposing that that setvar does not occur in the already defined 𝜑 is not a big constraint. (Contributed by BJ, 21-Oct-2019.)

𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 
TheoremstrcollnfALT 15483* Alternate proof of strcollnf 15482, not using strcollnft 15481. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 
13.2.13  CZF: Subset collection

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

 
Axiomax-sscoll 15484* Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function from 𝑎 to 𝑏, or equivalently a collection of nonempty subsets of 𝑏 indexed by 𝑎, and the consequent asserts the existence of a subset of 𝑐 which "collects" at least one element in the image of each 𝑥𝑎 and which is made only of such elements. The axiom asserts the existence, for any sets 𝑎, 𝑏, of a set 𝑐 such that that implication holds for any value of the parameter 𝑧 of 𝜑. (Contributed by BJ, 5-Oct-2019.)
𝑎𝑏𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐 (∀𝑥𝑎𝑦𝑑 𝜑 ∧ ∀𝑦𝑑𝑥𝑎 𝜑))
 
Theoremsscoll2 15485* Version of ax-sscoll 15484 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐 (∀𝑥𝑎𝑦𝑑 𝜑 ∧ ∀𝑦𝑑𝑥𝑎 𝜑))
 
13.2.14  Real numbers
 
Axiomax-ddkcomp 15486 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 Axiom ax-ddkcomp 15486 should be used in place of construction specific results. In particular, axcaucvg 7960 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
(((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧𝐴 𝑥 < 𝑧 ∨ ∀𝑧𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 𝑦𝑥 ∧ ((𝐵𝑅 ∧ ∀𝑦𝐴 𝑦𝐵) → 𝑥𝐵)))
 
13.3  Mathbox for Jim Kingdon
 
13.3.1  Propositional and predicate logic
 
Theoremnnnotnotr 15487 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 851, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
¬ ¬ (¬ ¬ 𝜑𝜑)
 
13.3.2  Natural numbers
 
Theorem1dom1el 15488 If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
((𝐴 ≼ 1o𝐵𝐴𝐶𝐴) → 𝐵 = 𝐶)
 
Theoremss1oel2o 15489 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4227 which more directly illustrates the contrast with el2oss1o 6496. (Contributed by Jim Kingdon, 8-Aug-2022.)
(EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))
 
Theoremnnti 15490 Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.)
(𝜑𝐴 ∈ ω)       ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢)))
 
Theorem012of 15491 Mapping zero and one between 0 and ω style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐺 ↾ {0, 1}):{0, 1}⟶2o
 
Theorem2o01f 15492 Mapping zero and one between ω and 0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)       (𝐺 ↾ 2o):2o⟶{0, 1}
 
13.3.3  The power set of a singleton
 
Theorempwtrufal 15493 A subset of the singleton {∅} cannot be anything other than or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4227. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4225), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
(𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅}))
 
Theorempwle2 15494* An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = 𝑥𝑁 ({𝑥} × 1o)       ((𝑁 ∈ ω ∧ 𝐺:𝑇1-1→𝒫 1o) → 𝑁 ⊆ 2o)
 
Theorempwf1oexmid 15495* An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
𝑇 = 𝑥𝑁 ({𝑥} × 1o)       ((𝑁 ∈ ω ∧ 𝐺:𝑇1-1→𝒫 1o) → (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2oEXMID)))
 
Theoremsubctctexmid 15496* If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o)))    &   (𝜑 → ω ∈ Markov)       (𝜑EXMID)
 
Theoremsssneq 15497* Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
(𝐴 ⊆ {𝐵} → ∀𝑦𝐴𝑧𝐴 𝑦 = 𝑧)
 
Theorempw1nct 15498* A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
(∀𝑟(𝑟 ⊆ (𝒫 1o × ω) → (∀𝑝 ∈ 𝒫 1o𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1o ⊔ 1o))
 
13.3.4  Omniscience of NN+oo
 
Theorem0nninf 15499 The zero element of (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.)
(ω × {∅}) ∈ ℕ
 
Theoremnnsf 15500* Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       𝑆:ℕ⟶ℕ
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