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Theorem List for Intuitionistic Logic Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-prexg 12801 Proof of prexg 4093 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)

Theorembj-snexg 12802 snexg 4068 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝐴} ∈ V)

Theorembj-snex 12803 snex 4069 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       {𝐴} ∈ V

Theorembj-sels 12804* 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 12805* axun2 4317 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))

Theorembj-uniex2 12806* uniex2 4318 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
𝑦 𝑦 = 𝑥

Theorembj-uniex 12807 uniex 4319 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V        𝐴 ∈ V

Theorembj-uniexg 12808 uniexg 4321 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 𝐴 ∈ V)

Theorembj-unex 12809 unex 4322 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V

Theorembdunexb 12810 Bounded version of unexb 4323. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝐵       ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Theorembj-unexg 12811 unexg 4324 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theorembj-sucexg 12812 sucexg 4374 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → suc 𝐴 ∈ V)

Theorembj-sucex 12813 sucex 4375 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
𝐴 ∈ V       suc 𝐴 ∈ V

10.2.7.1  Delta_0-classical logic

Axiomax-bj-d0cl 12814 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
BOUNDED 𝜑       DECID 𝜑

Theorembj-d0clsepcl 12815 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
DECID 𝜑

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

Syntaxwind 12816 Syntax for inductive classes.
wff Ind 𝐴

Definitiondf-bj-ind 12817* Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))

Theorembj-indsuc 12818 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
(Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))

Theorembj-indeq 12819 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
(𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))

Theorembj-bdind 12820 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
BOUNDED Ind 𝑥

Theorembj-indint 12821* The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Ind {𝑥𝐴 ∣ Ind 𝑥}

Theorembj-indind 12822* 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 12823 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 12824 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
Ind ω

Theorembj-omssind 12825 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 12826* A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∀𝑥(Ind 𝑥𝐴𝑥) ↔ 𝐴 ⊆ ω)

Theorembj-om 12827* 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 12828* Two formulations of the axiom of infinity (see ax-infvn 12831 and bj-omex 12832) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦)))

10.2.7.3  The first three Peano postulates

The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4468 and peano3 4470 already show this. In this section, we prove bj-peano2 12829 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms.

Theorembj-peano2 12829 Constructive proof of peano2 4469. Temporary note: another possibility is to simply replace sucexg 4374 with bj-sucexg 12812 in the proof of peano2 4469. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)

Theorempeano5set 12830* Version of peano5 4472 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))

10.2.8  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.

10.2.8.1  The set of natural numbers (finite ordinals)

In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 12831) and deduce that the class ω of finite ordinals is a set (bj-omex 12832).

Axiomax-infvn 12831* 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 4462) from which one then proves, using full separation, that the wanted set exists (omex 4467). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))

Theorembj-omex 12832 Proof of omex 4467 from ax-infvn 12831. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.)
ω ∈ V

10.2.8.2  Peano's fifth postulate

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

Theorembdpeano5 12833* Bounded version of peano5 4472. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴       ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)

Theoremspeano5 12834* Version of peano5 4472 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)

10.2.8.3  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 12835* Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4473 for a nonconstructive proof of the general case. See bdfind 12836 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 12836* Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4473 for a nonconstructive proof of the general case. See findset 12835 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 12837* 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 4474 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4474, finds2 4475, finds1 4476. (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 12838* Version of bj-bdfindis 12837 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 12837 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 12839 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 12837 for explanations. From this version, it is easy to prove the bounded version of findes 4477. (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 12840* Constructive proof of a variant of nn0suc 4478. For a constructive proof of nn0suc 4478, see bj-nn0suc 12854. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))

Theorembj-nntrans 12841 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → (𝐵𝐴𝐵𝐴))

Theorembj-nntrans2 12842 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Tr 𝐴)

Theorembj-nnelirr 12843 A natural number does not belong to itself. Version of elirr 4416 for natural numbers, which does not require ax-setind 4412. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
(𝐴 ∈ ω → ¬ 𝐴𝐴)

Theorembj-nnen2lp 12844 A version of en2lp 4429 for natural numbers, which does not require ax-setind 4412.

Note: using this theorem and bj-nnelirr 12843, one can remove dependency on ax-setind 4412 from nntri2 6344 and nndcel 6350; 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 12845 Remove from peano4 4471 dependency on ax-setind 4412. 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 12846 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4479.

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 12847 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Tr ω

Theorembj-nnord 12848 A natural number is an ordinal. Constructive proof of nnord 4485. Can also be proved from bj-nnelon 12849 if the latter is proved from bj-omssonALT 12853. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)

Theorembj-nnelon 12849 A natural number is an ordinal. Constructive proof of nnon 4483. Can also be proved from bj-omssonALT 12853. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)

Theorembj-omord 12850 The set ω is an ordinal. Constructive proof of ordom 4480. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
Ord ω

Theorembj-omelon 12851 The set ω is an ordinal. Constructive proof of omelon 4482. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
ω ∈ On

Theorembj-omsson 12852 Constructive proof of omsson 4486. See also bj-omssonALT 12853. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On

Theorembj-omssonALT 12853 Alternate proof of bj-omsson 12852. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On

Theorembj-nn0suc 12854* Proof of (biconditional form of) nn0suc 4478 from the core axioms of CZF. See also bj-nn0sucALT 12868. 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))

10.2.9  CZF: Set induction

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

10.2.9.1  Set induction

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

Theoremsetindft 12855* 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 12856* 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 12857* 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 12858* Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
BOUNDED 𝜑       (∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)

Theorembdsetindis 12859* 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 12860* Lemma for bj-inf2vn 12864. 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 12861* Lemma for bj-inf2vnlem3 12862 and bj-inf2vnlem4 12863. 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 12862* Lemma for bj-inf2vn 12864. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
BOUNDED 𝐴    &   BOUNDED 𝑍       (∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))

Theorembj-inf2vnlem4 12863* Lemma for bj-inf2vn2 12865. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥𝐴 (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦) → (Ind 𝑍𝐴𝑍))

Theorembj-inf2vn 12864* A sufficient condition for ω to be a set. See bj-inf2vn2 12865 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 12865* A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 12864. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
(𝐴𝑉 → (∀𝑥(𝑥𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))

Axiomax-inf2 12866* Another axiom of infinity in a constructive setting (see ax-infvn 12831). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
𝑎𝑥(𝑥𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦𝑎 𝑥 = suc 𝑦))

Theorembj-omex2 12867 Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 12831 (see bj-2inf 12828 for the equivalence of the latter with bj-omex 12832). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ∈ V

Theorembj-nn0sucALT 12868* Alternate proof of bj-nn0suc 12854, also constructive but from ax-inf2 12866, hence requiring ax-bdsetind 12858. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

10.2.9.2  Full induction

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

Theorembj-findis 12869* 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 12837 for a bounded version not requiring ax-setind 4412. See finds 4474 for a proof in IZF. From this version, it is easy to prove of finds 4474, finds2 4475, finds1 4476. (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 12870* Version of bj-findis 12869 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 12869 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 12871 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 12869 for explanations. From this version, it is easy to prove findes 4477. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
(([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

10.2.10  CZF: Strong collection

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

Axiomax-strcoll 12872* 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 12873* Version of ax-strcoll 12872 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
(∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

Theoremstrcollnft 12874* Closed form of strcollnf 12875. Version of ax-strcoll 12872 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 12875* Version of ax-strcoll 12872 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 12876* Alternate proof of strcollnf 12875, not using strcollnft 12874. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))

10.2.11  CZF: Subset collection

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

Axiomax-sscoll 12877* 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 12878* Version of ax-sscoll 12877 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.)
𝑐𝑧(∀𝑥𝑎𝑦𝑏 𝜑 → ∃𝑑𝑐𝑦(𝑦𝑑 ↔ ∃𝑥𝑎 𝜑))

10.2.12  Real numbers

Axiomax-ddkcomp 12879 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 the axiom ax-ddkcomp 12879 should be used in place of construction specific results. In particular, axcaucvg 7635 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)))

10.3  Mathbox for Jim Kingdon

10.3.1  Natural numbers

Theoremel2oss1o 12880 Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 12881. (Contributed by Jim Kingdon, 8-Aug-2022.)
(𝐴 ∈ 2o𝐴 ⊆ 1o)

Theoremss1oel2o 12881 Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4081 which more directly illustrates the contrast with el2oss1o 12880. (Contributed by Jim Kingdon, 8-Aug-2022.)
(EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))

Theorempw1dom2 12882 The power set of 1o dominates 2o. Also see pwpw0ss 3697 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
2o ≼ 𝒫 1o

Theoremnnti 12883 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)))

10.3.2  The power set of a singleton

Theorempwtrufal 12884 A subset of the singleton {∅} cannot be anything other than or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4081. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4080), 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 12885* 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 12886* 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)))

Theoremexmid1stab 12887* If any proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
((𝜑𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})       (𝜑EXMID)

Theoremsubctctexmid 12888* 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)

10.3.3  Omniscience of NN+oo

Theorem0nninf 12889 The zero element of (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.)
(ω × {∅}) ∈ ℕ

Theoremnninff 12890 An element of is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝐴 ∈ ℕ𝐴:ω⟶2o)

Theoremnnsf 12891* 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

Theorempeano4nninf 12892* The successor function on is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       𝑆:ℕ1-1→ℕ

Theorempeano3nninf 12893* The successor function on is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
𝑆 = (𝑝 ∈ ℕ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝 𝑖))))       (𝐴 ∈ ℕ → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Theoremnninfalllemn 12894* Lemma for nninfall 12896. Mapping of a natural number to an element of . (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑁 ∈ ω)    &   (𝜑 → ∀𝑥𝑁 (𝑃𝑥) = 1o)    &   (𝜑 → (𝑃𝑁) = ∅)       (𝜑𝑃 = (𝑖 ∈ ω ↦ if(𝑖𝑁, 1o, ∅)))

Theoremnninfalllem1 12895* Lemma for nninfall 12896. (Contributed by Jim Kingdon, 1-Aug-2022.)
(𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)    &   (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑 → (𝑄𝑃) = ∅)       (𝜑 → ∀𝑛 ∈ ω (𝑃𝑛) = 1o)

Theoremnninfall 12896* Given a decidable predicate on , showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which 𝑄 is a decidable predicate is that it assigns a value of either or 1o (which can be thought of as false and true) to every element of . Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)
(𝜑𝑄 ∈ (2o𝑚))    &   (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o)    &   (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) = 1o)       (𝜑 → ∀𝑝 ∈ ℕ (𝑄𝑝) = 1o)

Theoremnninfex 12897 is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
∈ V

Theoremnninfsellemdc 12898* Lemma for nninfself 12901. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)
((𝑄 ∈ (2o𝑚) ∧ 𝑁 ∈ ω) → DECID𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o)

Theoremnninfsellemcl 12899* Lemma for nninfself 12901. (Contributed by Jim Kingdon, 8-Aug-2022.)
((𝑄 ∈ (2o𝑚) ∧ 𝑁 ∈ ω) → if(∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o)

Theoremnninfsellemsuc 12900* Lemma for nninfself 12901. (Contributed by Jim Kingdon, 6-Aug-2022.)
((𝑄 ∈ (2o𝑚) ∧ 𝐽 ∈ ω) → if(∀𝑘 ∈ suc suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅) ⊆ if(∀𝑘 ∈ suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖𝑘, 1o, ∅))) = 1o, 1o, ∅))

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