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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-indind 13301* | 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 13302 | 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 13303 | ω is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
⊢ Ind ω | ||
Theorem | bj-omssind 13304 | 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 13305* | 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 13306* | 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 13307* | Two formulations of the axiom of infinity (see ax-infvn 13310 and bj-omex 13311) . (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 4516 and peano3 4518 already show this. In this section, we prove bj-peano2 13308 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms. | ||
Theorem | bj-peano2 13308 | Constructive proof of peano2 4517. Temporary note: another possibility is to simply replace sucexg 4422 with bj-sucexg 13291 in the proof of peano2 4517. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) | ||
Theorem | peano5set 13309* | Version of peano5 4520 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 13310) and deduce that the class ω of finite ordinals is a set (bj-omex 13311). | ||
Axiom | ax-infvn 13310* | 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 4510) from which one then proves, using full separation, that the wanted set exists (omex 4515). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.) |
⊢ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)) | ||
Theorem | bj-omex 13311 | Proof of omex 4515 from ax-infvn 13310. (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 13312* | Bounded version of peano5 4520. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ BOUNDED 𝐴 ⇒ ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | speano5 13313* | Version of peano5 4520 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 finite ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers. | ||
Theorem | findset 13314* | Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4521 for a nonconstructive proof of the general case. See bdfind 13315 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 13315* | Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4521 for a nonconstructive proof of the general case. See findset 13314 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 13316* | 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 4522 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4522, finds2 4523, finds1 4524. (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 13317* | Version of bj-bdfindis 13316 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13316 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 13318 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13316 for explanations. From this version, it is easy to prove the bounded version of findes 4525. (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 13319* | Constructive proof of a variant of nn0suc 4526. For a constructive proof of nn0suc 4526, see bj-nn0suc 13333. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥)) | ||
Theorem | bj-nntrans 13320 | 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 13321 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → Tr 𝐴) | ||
Theorem | bj-nnelirr 13322 | A natural number does not belong to itself. Version of elirr 4464 for natural numbers, which does not require ax-setind 4460. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | bj-nnen2lp 13323 |
A version of en2lp 4477 for natural numbers, which does not require
ax-setind 4460.
Note: using this theorem and bj-nnelirr 13322, one can remove dependency on ax-setind 4460 from nntri2 6398 and nndcel 6404; 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 13324 | Remove from peano4 4519 dependency on ax-setind 4460. 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 13325 |
The set ω is transitive. A natural number is
included in
ω. Constructive proof of elnn 4527.
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 13326 | The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Tr ω | ||
Theorem | bj-nnord 13327 | A natural number is an ordinal. Constructive proof of nnord 4533. Can also be proved from bj-nnelon 13328 if the latter is proved from bj-omssonALT 13332. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | bj-nnelon 13328 | A natural number is an ordinal. Constructive proof of nnon 4531. Can also be proved from bj-omssonALT 13332. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | bj-omord 13329 | The set ω is an ordinal. Constructive proof of ordom 4528. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ Ord ω | ||
Theorem | bj-omelon 13330 | The set ω is an ordinal. Constructive proof of omelon 4530. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
⊢ ω ∈ On | ||
Theorem | bj-omsson 13331 | Constructive proof of omsson 4534. See also bj-omssonALT 13332. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
⊢ ω ⊆ On | ||
Theorem | bj-omssonALT 13332 | Alternate proof of bj-omsson 13331. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ω ⊆ On | ||
Theorem | bj-nn0suc 13333* | Proof of (biconditional form of) nn0suc 4526 from the core axioms of CZF. See also bj-nn0sucALT 13347. 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 13334* | Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)) | ||
Theorem | setindf 13335* | Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) | ||
Theorem | setindis 13336* | 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 13337* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
⊢ BOUNDED 𝜑 ⇒ ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑) | ||
Theorem | bdsetindis 13338* | 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 13339* | Lemma for bj-inf2vn 13343. 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 13340* | Lemma for bj-inf2vnlem3 13341 and bj-inf2vnlem4 13342. 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 13341* | Lemma for bj-inf2vn 13343. (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 13342* | Lemma for bj-inf2vn2 13344. (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 13343* | A sufficient condition for ω to be a set. See bj-inf2vn2 13344 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 13344* | A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 13343. (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 13345* | Another axiom of infinity in a constructive setting (see ax-infvn 13310). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦)) | ||
Theorem | bj-omex2 13346 | Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 13310 (see bj-2inf 13307 for the equivalence of the latter with bj-omex 13311). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ω ∈ V | ||
Theorem | bj-nn0sucALT 13347* | Alternate proof of bj-nn0suc 13333, also constructive but from ax-inf2 13345, hence requiring ax-bdsetind 13337. (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 13348* | 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 13316 for a bounded version not requiring ax-setind 4460. See finds 4522 for a proof in IZF. From this version, it is easy to prove of finds 4522, finds2 4523, finds1 4524. (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 13349* | Version of bj-findis 13348 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13348 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 13350 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13348 for explanations. From this version, it is easy to prove findes 4525. (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 13351* | 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 4051. (Contributed by BJ, 5-Oct-2019.) |
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
Theorem | strcoll2 13352* | Version of ax-strcoll 13351 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 13353* | Closed form of strcollnf 13354. (Contributed by BJ, 21-Oct-2019.) |
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) | ||
Theorem | strcollnf 13354* |
Version of ax-strcoll 13351 with one disjoint variable condition
removed,
the other disjoint variable condition replaced with a non-freeness
hypothesis, and without initial universal quantifier. Version of
strcoll2 13352 with the disjoint variable condition on
𝑏, 𝜑 replaced
with a non-freeness hypothesis.
This proof aims to demonstrate a standard technique, but strcoll2 13352 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 13355* | Alternate proof of strcollnf 13354, not using strcollnft 13353. (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 13356* | 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 13357* | Version of ax-sscoll 13356 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 13358 | 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 13358 should be used in place of construction specific results. In particular, axcaucvg 7732 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 | el2oss1o 13359 | Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13360. (Contributed by Jim Kingdon, 8-Aug-2022.) |
⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o) | ||
Theorem | ss1oel2o 13360 | Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4129 which more directly illustrates the contrast with el2oss1o 13359. (Contributed by Jim Kingdon, 8-Aug-2022.) |
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) | ||
Theorem | pw1dom2 13361 | The power set of 1o dominates 2o. Also see pwpw0ss 3739 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.) |
⊢ 2o ≼ 𝒫 1o | ||
Theorem | nnti 13362 | 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 13363 | 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 13364 | 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 13365 | A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4129. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4128), 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 13366* | 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 13367* | 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 13368* | 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 13369* | 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 13370* | 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 13371* | 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 13372 | The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.) |
⊢ (ω × {∅}) ∈ ℕ∞ | ||
Theorem | nninff 13373 | An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o) | ||
Theorem | nnsf 13374* | 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∞ | ||
Theorem | peano4nninf 13375* | The successor function on ℕ∞ 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:ℕ∞–1-1→ℕ∞ | ||
Theorem | peano3nninf 13376* | The successor function on ℕ∞ 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∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅)) | ||
Theorem | nninfalllemn 13377* | Lemma for nninfall 13379. Mapping of a natural number to an element of ℕ∞. (Contributed by Jim Kingdon, 4-Aug-2022.) |
⊢ (𝜑 → 𝑃 ∈ ℕ∞) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 (𝑃‘𝑥) = 1o) & ⊢ (𝜑 → (𝑃‘𝑁) = ∅) ⇒ ⊢ (𝜑 → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) | ||
Theorem | nninfalllem1 13378* | Lemma for nninfall 13379. (Contributed by Jim Kingdon, 1-Aug-2022.) |
⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o) & ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o) & ⊢ (𝜑 → 𝑃 ∈ ℕ∞) & ⊢ (𝜑 → (𝑄‘𝑃) = ∅) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 1o) | ||
Theorem | nninfall 13379* | Given a decidable predicate on ℕ∞, 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 ℕ∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o) & ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o) ⇒ ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o) | ||
Theorem | nninfex 13380 | ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
⊢ ℕ∞ ∈ V | ||
Theorem | nninfsellemdc 13381* | Lemma for nninfself 13384. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.) |
⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o) | ||
Theorem | nninfsellemcl 13382* | Lemma for nninfself 13384. (Contributed by Jim Kingdon, 8-Aug-2022.) |
⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → if(∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o) | ||
Theorem | nninfsellemsuc 13383* | Lemma for nninfself 13384. (Contributed by Jim Kingdon, 6-Aug-2022.) |
⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝐽 ∈ ω) → if(∀𝑘 ∈ suc suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ⊆ if(∀𝑘 ∈ suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) | ||
Theorem | nninfself 13384* | Domain and range of the selection function for ℕ∞. (Contributed by Jim Kingdon, 6-Aug-2022.) |
⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) ⇒ ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞ | ||
Theorem | nninfsellemeq 13385* | Lemma for nninfsel 13388. (Contributed by Jim Kingdon, 9-Aug-2022.) |
⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → ∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o) & ⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = ∅) ⇒ ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) | ||
Theorem | nninfsellemqall 13386* | Lemma for nninfsel 13388. (Contributed by Jim Kingdon, 9-Aug-2022.) |
⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) & ⊢ (𝜑 → 𝑁 ∈ ω) ⇒ ⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = 1o) | ||
Theorem | nninfsellemeqinf 13387* | Lemma for nninfsel 13388. (Contributed by Jim Kingdon, 9-Aug-2022.) |
⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) ⇒ ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o)) | ||
Theorem | nninfsel 13388* | 𝐸 is a selection function for ℕ∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.) |
⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) ⇒ ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o) | ||
Theorem | nninfomnilem 13389* | Lemma for nninfomni 13390. (Contributed by Jim Kingdon, 10-Aug-2022.) |
⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) ⇒ ⊢ ℕ∞ ∈ Omni | ||
Theorem | nninfomni 13390 | ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.) |
⊢ ℕ∞ ∈ Omni | ||
Theorem | nninffeq 13391* | Equality of two functions on ℕ∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 4-Aug-2023.) |
⊢ (𝜑 → 𝐹:ℕ∞⟶ℕ0) & ⊢ (𝜑 → 𝐺:ℕ∞⟶ℕ0) & ⊢ (𝜑 → (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑥 ∈ ω ↦ 1o))) & ⊢ (𝜑 → ∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | exmidsbthrlem 13392* | Lemma for exmidsbthr 13393. (Contributed by Jim Kingdon, 11-Aug-2022.) |
⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))) ⇒ ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) | ||
Theorem | exmidsbthr 13393* | The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.) |
⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) | ||
Theorem | exmidsbth 13394* |
The Schroeder-Bernstein Theorem is equivalent to excluded middle. This
is Metamath 100 proof #25. The forward direction (isbth 6863) is the
proof of the Schroeder-Bernstein Theorem from the Metamath Proof
Explorer database (in which excluded middle holds), but adapted to use
EXMID as an antecedent rather
than being unconditionally true, as in
the non-intuitionist proof at
https://us.metamath.org/mpeuni/sbth.html 6863.
The reverse direction (exmidsbthr 13393) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.) |
⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) | ||
Theorem | sbthomlem 13395 | Lemma for sbthom 13396. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
⊢ (𝜑 → ω ∈ Omni) & ⊢ (𝜑 → 𝑌 ⊆ {∅}) & ⊢ (𝜑 → 𝐹:ω–1-1-onto→(𝑌 ⊔ ω)) ⇒ ⊢ (𝜑 → (𝑌 = ∅ ∨ 𝑌 = {∅})) | ||
Theorem | sbthom 13396 | Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 13394 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ω? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.) |
⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID) | ||
Theorem | qdencn 13397* | The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 11006 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.) |
⊢ 𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)} ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ 𝑄 (abs‘(𝑥 − 𝐴)) < 𝐵) | ||
Theorem | refeq 13398* | Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.) |
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥))) & ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | triap 13399 | Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ DECID 𝐴 # 𝐵)) | ||
Theorem | isomninnlem 13400* | Lemma for isomninn 13401. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.) |
⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))) |
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