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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bdssexg 13901 | Bounded version of ssexg 4126. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | bdssexd 13902 | Bounded version of ssexd 4127. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝜑 → 𝐴 ∈ V) | ||
Theorem | bdrabexg 13903* | Bounded version of rabexg 4130. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ BOUNDED 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | bj-inex 13904 | The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | bj-intexr 13905 | intexr 4134 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) | ||
Theorem | bj-intnexr 13906 | intnexr 4135 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) | ||
Theorem | bj-zfpair2 13907 | Proof of zfpair2 4193 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | bj-prexg 13908 | Proof of prexg 4194 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | ||
Theorem | bj-snexg 13909 | snexg 4168 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | ||
Theorem | bj-snex 13910 | snex 4169 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ∈ V | ||
Theorem | bj-sels 13911* | If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) | ||
Theorem | bj-axun2 13912* | axun2 4418 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) | ||
Theorem | bj-uniex2 13913* | uniex2 4419 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
⊢ ∃𝑦 𝑦 = ∪ 𝑥 | ||
Theorem | bj-uniex 13914 | uniex 4420 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∪ 𝐴 ∈ V | ||
Theorem | bj-uniexg 13915 | uniexg 4422 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | ||
Theorem | bj-unex 13916 | unex 4424 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∪ 𝐵) ∈ V | ||
Theorem | bdunexb 13917 | Bounded version of unexb 4425. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝐵 ⇒ ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | bj-unexg 13918 | unexg 4426 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | bj-sucexg 13919 | sucexg 4480 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | ||
Theorem | bj-sucex 13920 | sucex 4481 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ suc 𝐴 ∈ V | ||
Axiom | ax-bj-d0cl 13921 | Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ DECID 𝜑 | ||
Theorem | bj-d0clsepcl 13922 | Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
⊢ DECID 𝜑 | ||
Syntax | wind 13923 | Syntax for inductive classes. |
wff Ind 𝐴 | ||
Definition | df-bj-ind 13924* | Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) |
⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | ||
Theorem | bj-indsuc 13925 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) | ||
Theorem | bj-indeq 13926 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵)) | ||
Theorem | bj-bdind 13927 | Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
⊢ BOUNDED Ind 𝑥 | ||
Theorem | bj-indint 13928* | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} | ||
Theorem | bj-indind 13929* | If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴 ∩ 𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.) |
⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵)) | ||
Theorem | bj-dfom 13930 | 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} | ||
Theorem | bj-omind 13931 | ω is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind ω | ||
Theorem | bj-omssind 13932 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴)) | ||
Theorem | bj-ssom 13933* | A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω) | ||
Theorem | bj-om 13934* | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥)))) | ||
Theorem | bj-2inf 13935* | Two formulations of the axiom of infinity (see ax-infvn 13938 and bj-omex 13939) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
⊢ (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦))) | ||
The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4576 and peano3 4578 already show this. In this section, we prove bj-peano2 13936 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms. | ||
Theorem | bj-peano2 13936 | Constructive proof of peano2 4577. Temporary note: another possibility is to simply replace sucexg 4480 with bj-sucexg 13919 in the proof of peano2 4577. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) | ||
Theorem | peano5set 13937* | Version of peano5 4580 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 (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)) | ||
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. | ||
In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13938) and deduce that the class ω of natural number ordinals is a set (bj-omex 13939). | ||
Axiom | ax-infvn 13938* | 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 4570) from which one then proves, using full separation, that the wanted set exists (omex 4575). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.) |
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) | ||
Theorem | bj-omex 13939 | Proof of omex 4575 from ax-infvn 13938. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
⊢ ω ∈ V | ||
In this section, we give constructive proofs of two versions of Peano's fifth postulate. | ||
Theorem | bdpeano5 13940* | Bounded version of peano5 4580. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | speano5 13941* | Version of peano5 4580 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
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. | ||
Theorem | findset 13942* | Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4581 for a nonconstructive proof of the general case. See bdfind 13943 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)) | ||
Theorem | bdfind 13943* | Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4581 for a nonconstructive proof of the general case. See findset 13942 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 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω) | ||
Theorem | bj-bdfindis 13944* | 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 4582 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4582, finds2 4583, finds1 4584. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) | ||
Theorem | bj-bdfindisg 13945* | Version of bj-bdfindis 13944 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13944 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜏 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) | ||
Theorem | bj-bdfindes 13946 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13944 for explanations. From this version, it is easy to prove the bounded version of findes 4585. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) | ||
Theorem | bj-nn0suc0 13947* | Constructive proof of a variant of nn0suc 4586. For a constructive proof of nn0suc 4586, see bj-nn0suc 13961. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥)) | ||
Theorem | bj-nntrans 13948 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | ||
Theorem | bj-nntrans2 13949 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → Tr 𝐴) | ||
Theorem | bj-nnelirr 13950 | A natural number does not belong to itself. Version of elirr 4523 for natural numbers, which does not require ax-setind 4519. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | bj-nnen2lp 13951 |
A version of en2lp 4536 for natural numbers, which does not require
ax-setind 4519.
Note: using this theorem and bj-nnelirr 13950, one can remove dependency on ax-setind 4519 from nntri2 6471 and nndcel 6477; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | ||
Theorem | bj-peano4 13952 | Remove from peano4 4579 dependency on ax-setind 4519. 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 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-omtrans 13953 |
The set ω is transitive. A natural number is
included in
ω. Constructive proof of elnn 4588.
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 𝑥 ⊆ 𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | ||
Theorem | bj-omtrans2 13954 | The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Tr ω | ||
Theorem | bj-nnord 13955 | A natural number is an ordinal class. Constructive proof of nnord 4594. Can also be proved from bj-nnelon 13956 if the latter is proved from bj-omssonALT 13960. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | bj-nnelon 13956 | A natural number is an ordinal. Constructive proof of nnon 4592. Can also be proved from bj-omssonALT 13960. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | bj-omord 13957 | The set ω is an ordinal class. Constructive proof of ordom 4589. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Ord ω | ||
Theorem | bj-omelon 13958 | The set ω is an ordinal. Constructive proof of omelon 4591. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ ω ∈ On | ||
Theorem | bj-omsson 13959 | Constructive proof of omsson 4595. See also bj-omssonALT 13960. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
⊢ ω ⊆ On | ||
Theorem | bj-omssonALT 13960 | Alternate proof of bj-omsson 13959. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ω ⊆ On | ||
Theorem | bj-nn0suc 13961* | Proof of (biconditional form of) nn0suc 4586 from the core axioms of CZF. See also bj-nn0sucALT 13975. 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)) | ||
In this section, we add the axiom of set induction to the core axioms of CZF. | ||
In this section, we prove some variants of the axiom of set induction. | ||
Theorem | setindft 13962* | Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)) | ||
Theorem | setindf 13963* | Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) | ||
Theorem | setindis 13964* | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑)) ⇒ ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑) | ||
Axiom | ax-bdsetind 13965* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑) | ||
Theorem | bdsetindis 13966* | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑)) ⇒ ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑) | ||
Theorem | bj-inf2vnlem1 13967* | Lemma for bj-inf2vn 13971. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴) | ||
Theorem | bj-inf2vnlem2 13968* | Lemma for bj-inf2vnlem3 13969 and bj-inf2vnlem4 13970. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍)))) | ||
Theorem | bj-inf2vnlem3 13969* | Lemma for bj-inf2vn 13971. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 & ⊢ BOUNDED 𝑍 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍)) | ||
Theorem | bj-inf2vnlem4 13970* | Lemma for bj-inf2vn2 13972. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍)) | ||
Theorem | bj-inf2vn 13971* | A sufficient condition for ω to be a set. See bj-inf2vn2 13972 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω)) | ||
Theorem | bj-inf2vn2 13972* | A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13971. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω)) | ||
Axiom | ax-inf2 13973* | Another axiom of infinity in a constructive setting (see ax-infvn 13938). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) | ||
Theorem | bj-omex2 13974 | Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13938 (see bj-2inf 13935 for the equivalence of the latter with bj-omex 13939). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ω ∈ V | ||
Theorem | bj-nn0sucALT 13975* | Alternate proof of bj-nn0suc 13961, also constructive but from ax-inf2 13973, hence requiring ax-bdsetind 13965. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) | ||
In this section, using the axiom of set induction, we prove full induction on the set of natural numbers. | ||
Theorem | bj-findis 13976* | 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 13944 for a bounded version not requiring ax-setind 4519. See finds 4582 for a proof in IZF. From this version, it is easy to prove of finds 4582, finds2 4583, finds1 4584. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) | ||
Theorem | bj-findisg 13977* | Version of bj-findis 13976 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13976 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜏 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) | ||
Theorem | bj-findes 13978 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13976 for explanations. From this version, it is easy to prove findes 4585. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) | ||
In this section, we state the axiom scheme of strong collection, which is part of CZF set theory. | ||
Axiom | ax-strcoll 13979* | 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 𝑥 ∈ 𝑎 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 4102. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcoll2 13980* | Version of ax-strcoll 13979 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcollnft 13981* | Closed form of strcollnf 13982. (Contributed by BJ, 21-Oct-2019.) |
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) | ||
Theorem | strcollnf 13982* |
Version of ax-strcoll 13979 with one disjoint variable condition
removed,
the other disjoint variable condition replaced with a nonfreeness
hypothesis, and without initial universal quantifier. Version of
strcoll2 13980 with the disjoint variable condition on
𝑏, 𝜑 replaced
with a nonfreeness hypothesis.
This proof aims to demonstrate a standard technique, but strcoll2 13980 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.) |
⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcollnfALT 13983* | Alternate proof of strcollnf 13982, not using strcollnft 13981. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
In this section, we state the axiom scheme of subset collection, which is part of CZF set theory. | ||
Axiom | ax-sscoll 13984* | 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 𝑥 ∈ 𝑎 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.) |
⊢ ∀𝑎∀𝑏∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | sscoll2 13985* | Version of ax-sscoll 13984 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Axiom | ax-ddkcomp 13986 | 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 13986 should be used in place of construction specific results. In particular, axcaucvg 7851 should be proved from it. (Contributed by BJ, 24-Oct-2021.) |
⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝐵) → 𝑥 ≤ 𝐵))) | ||
Theorem | nnnotnotr 13987 | Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 845, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.) |
⊢ ¬ ¬ (¬ ¬ 𝜑 → 𝜑) | ||
Theorem | ss1oel2o 13988 | Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4182 which more directly illustrates the contrast with el2oss1o 6420. (Contributed by Jim Kingdon, 8-Aug-2022.) |
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) | ||
Theorem | nnti 13989 | Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.) |
⊢ (𝜑 → 𝐴 ∈ ω) ⇒ ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢))) | ||
Theorem | 012of 13990 | Mapping zero and one between ℕ0 and ω style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (◡𝐺 ↾ {0, 1}):{0, 1}⟶2o | ||
Theorem | 2o01f 13991 | Mapping zero and one between ω and ℕ0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝐺 ↾ 2o):2o⟶{0, 1} | ||
Theorem | pwtrufal 13992 | A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4182. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4180), 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 = {∅})) | ||
Theorem | pwle2 13993* | 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 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) ⇒ ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o) | ||
Theorem | pwf1oexmid 13994* | 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 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) ⇒ ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2o ∧ EXMID))) | ||
Theorem | exmid1stab 13995* | 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.) |
⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅}) ⇒ ⊢ (𝜑 → EXMID) | ||
Theorem | subctctexmid 13996* | 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:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o))) & ⊢ (𝜑 → ω ∈ Markov) ⇒ ⊢ (𝜑 → EXMID) | ||
Theorem | sssneq 13997* | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧) | ||
Theorem | pw1nct 13998* | 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 1o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1o ⊔ 1o)) | ||
Theorem | 0nninf 13999 | The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.) |
⊢ (ω × {∅}) ∈ ℕ∞ | ||
Theorem | nnsf 14000* | 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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