P. 173 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 17201-17300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-nnelirr 17201	A natural number does not belong to itself. Version of elirr 4690 for natural numbers, which does not require ax-setind 4686. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)
Theorem	bj-nnen2lp 17202	A version of en2lp 4703 for natural numbers, which does not require ax-setind 4686. Note: using this theorem and bj-nnelirr 17201, one can remove dependency on ax-setind 4686 from nntri2 6772 and nndcel 6778; 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 17203	Remove from peano4 4746 dependency on ax-setind 4686. 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 17204	The set ω is transitive. A natural number is included in ω. Constructive proof of elomssom 4754. 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 17205	The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Tr ω
Theorem	bj-nnord 17206	A natural number is an ordinal class. Constructive proof of nnord 4761. Can also be proved from bj-nnelon 17207 if the latter is proved from bj-omssonALT 17211. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	bj-nnelon 17207	A natural number is an ordinal. Constructive proof of nnon 4759. Can also be proved from bj-omssonALT 17211. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	bj-omord 17208	The set ω is an ordinal class. Constructive proof of ordom 4756. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Ord ω
Theorem	bj-omelon 17209	The set ω is an ordinal. Constructive proof of omelon 4758. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ ω ∈ On
Theorem	bj-omsson 17210	Constructive proof of omsson 4762. See also bj-omssonALT 17211. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
⊢ ω ⊆ On
Theorem	bj-omssonALT 17211	Alternate proof of bj-omsson 17210. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ⊆ On
Theorem	bj-nn0suc 17212*	Proof of (biconditional form of) nn0suc 4753 from the core axioms of CZF. See also bj-nn0sucALT 17226. 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))
14.3.11  CZF: Set induction In this section, we add the axiom of set induction to the core axioms of CZF.
14.3.11.1  Set induction In this section, we prove some variants of the axiom of set induction.
Theorem	setindft 17213*	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 17214*	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 17215*	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 17216*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
Theorem	bdsetindis 17217*	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 17218*	Lemma for bj-inf2vn 17222. 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 17219*	Lemma for bj-inf2vnlem3 17220 and bj-inf2vnlem4 17221. 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 17220*	Lemma for bj-inf2vn 17222. (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 17221*	Lemma for bj-inf2vn2 17223. (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 17222*	A sufficient condition for ω to be a set. See bj-inf2vn2 17223 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 17223*	A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 17222. (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 17224*	Another axiom of infinity in a constructive setting (see ax-infvn 17189). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
Theorem	bj-omex2 17225	Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 17189 (see bj-2inf 17186 for the equivalence of the latter with bj-omex 17190). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	bj-nn0sucALT 17226*	Alternate proof of bj-nn0suc 17212, also constructive but from ax-inf2 17224, hence requiring ax-bdsetind 17216. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
14.3.11.2  Full induction In this section, using the axiom of set induction, we prove full induction on the set of natural numbers.
Theorem	bj-findis 17227*	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 17195 for a bounded version not requiring ax-setind 4686. See finds 4749 for a proof in IZF. From this version, it is easy to prove of finds 4749, finds2 4750, finds1 4751. (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 17228*	Version of bj-findis 17227 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 17227 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 17229	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 17227 for explanations. From this version, it is easy to prove findes 4752. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
14.3.12  CZF: Strong collection In this section, we state the axiom scheme of strong collection, which is part of CZF set theory.
Axiom	ax-strcoll 17230*	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 4248. (Contributed by BJ, 5-Oct-2019.)
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcoll2 17231*	Version of ax-strcoll 17230 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 17232*	Closed form of strcollnf 17233. (Contributed by BJ, 21-Oct-2019.)
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
Theorem	strcollnf 17233*	Version of ax-strcoll 17230 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 17231 with the disjoint variable condition on 𝑏, 𝜑 replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 17231 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 17234*	Alternate proof of strcollnf 17233, not using strcollnft 17232. (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))
14.3.13  CZF: Subset collection In this section, we state the axiom scheme of subset collection, which is part of CZF set theory.
Axiom	ax-sscoll 17235*	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 17236*	Version of ax-sscoll 17235 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))
14.3.14  Real numbers
Axiom	ax-ddkcomp 17237	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 17237 should be used in place of construction specific results. In particular, axcaucvg 8273 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)))
14.4  Mathbox for Jim Kingdon
14.4.1  Propositional and predicate logic
Theorem	nnnotnotr 17238	Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 862, 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)
14.4.2  The sizes of sets
Theorem	ss1oel2o 17239	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4337 which more directly illustrates the contrast with el2oss1o 6721. (Contributed by Jim Kingdon, 8-Aug-2022.)
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o))
Theorem	3dom 17240*	A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.)
⊢ (3o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧))
Theorem	pw1ndom3lem 17241	Lemma for pw1ndom3 17242. (Contributed by Jim Kingdon, 14-Feb-2026.)
⊢ (𝜑 → 𝑋 ∈ 𝒫 1o)    &   ⊢ (𝜑 → 𝑌 ∈ 𝒫 1o)    &   ⊢ (𝜑 → 𝑍 ∈ 𝒫 1o)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≠ 𝑍)    ⇒   ⊢ (𝜑 → 𝑋 = ∅)
Theorem	pw1ndom3 17242	The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4338). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
⊢ ¬ 3o ≼ 𝒫 1o
Theorem	pw1ninf 17243	The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7223), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7219. (Contributed by Jim Kingdon, 14-Feb-2026.)
⊢ ¬ ω ≼ 𝒫 1o
Theorem	nnti 17244	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 17245	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 17246	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	pw1map 17247*	Mapping between (𝒫 1o ↑𝑚 𝐴) and subsets of 𝐴. (Contributed by Jim Kingdon, 9-Jan-2026.)
⊢ 𝐹 = (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
Theorem	pw1mapen 17248	Equinumerosity of (𝒫 1o ↑𝑚 𝐴) and the set of subsets of 𝐴. (Contributed by Jim Kingdon, 10-Jan-2026.)
⊢ (𝐴 ∈ 𝑉 → (𝒫 1o ↑𝑚 𝐴) ≈ 𝒫 𝐴)
14.4.3  The power set of a singleton
Theorem	pwtrufal 17249	A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4337. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4335), 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 17250*	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 17251*	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	subctctexmid 17252*	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	domomsubct 17253*	A set dominated by ω is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.)
⊢ (𝐴 ≼ ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴))
Theorem	sssneq 17254*	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 17255*	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	rabid1o 17256*	Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)
⊢ ({𝑥 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)
Theorem	pw1dceq 17257*	The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
Theorem	exmidnotnotr 17258	Excluded middle is equivalent to double negation elimination. Read an element of 𝒫 1o as being a truth value and 𝑥 = 1o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromnotnotr 1497. (Contributed by Jim Kingdon, 22-Apr-2026.)
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
Theorem	exmidcon 17259*	Excluded middle is equivalent to the form of contraposition which removes negation. Read an element of 𝒫 1o as being a truth value and 𝑥 = 1o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromcon 1498. (Contributed by Jim Kingdon, 22-Apr-2026.)
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o((¬ 𝑦 = 1o → ¬ 𝑥 = 1o) → (𝑥 = 1o → 𝑦 = 1o)))
Theorem	exmidpeirce 17260*	Excluded middle is equivalent to Peirce's law. Read an element of 𝒫 1o as being a truth value and 𝑥 = 1o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfrompeirce 1499. (Contributed by Jim Kingdon, 23-Apr-2026.)
⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1o(((𝑥 = 1o → 𝑦 = 1o) → 𝑥 = 1o) → 𝑥 = 1o))
Theorem	stnot 17261*	A proposition is double negation stable if and only if it is equivalent to a negated proposition. Here by "proposition" we mean a subset of a singleton (which is a choice which allows us to quantify over them). Posed as an exercise online by Yannick Forster. (Contributed by Jim Kingdon, 24-Jul-2026.)
⊢ (𝐴 ∈ 𝒫 1o → ((¬ ¬ 𝐴 = 1o → 𝐴 = 1o) ↔ ∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o)))
14.4.4  Weak excluded middle
Syntax	wwem 17262	Formula for an abbreviation of weak excluded middle.
wff WEXMID
Definition	df-wexmid 17263	Weak excluded middle is the principle that any negated proposition is decidable. (Contributed by Jim Kingdon, 30-Jul-2026.)
⊢ (WEXMID ↔ ∀𝑝 ∈ 𝒫 1o(¬ 𝑝 = 1o ∨ ¬ ¬ 𝑝 = 1o))
Theorem	wexmiddc 17264	Weak excluded middle expressed using WEXMID implies decidability of a negated proposition. (Contributed by Jim Kingdon, 30-Jul-2026.)
⊢ (WEXMID → DECID ¬ 𝜑)
Theorem	wexmiddiffilem 17265*	Lemma for wexmiddiffi 17266. The reverse direction, using different notation. (Contributed by Jim Kingdon, 29-Jul-2026.)
⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
Theorem	wexmiddiffi 17266*	Being able to subtract an arbitrary set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7204. (Contributed by Jim Kingdon, 29-Jul-2026.)
⊢ (WEXMID ↔ ∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin))
Theorem	wexmiddifxylem 17267*	Lemma for wexmiddifxylem 17267. Showing weak excluded middle given a suitable finite set. (Contributed by Jim Kingdon, 1-Aug-2026.)
⊢ (({1o} ∖ {{𝑥 ∈ 1o ∣ 𝜑}}) ∈ Fin → DECID ¬ 𝜑)
Theorem	wexmiddifxy 17268*	Being able to subtract an arbitrary finite set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7204. (Contributed by Jim Kingdon, 1-Aug-2026.)
⊢ (WEXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))
14.4.5  Disjoint unions
Theorem	idjudju 17269*	An indexed disjoint union over two sets equals the disjoint union of those two sets. This connects our notation for an indexed disjoint union, ∪ 𝑥 ∈ 𝐶({𝑥} × 𝐹) which uses df-iun 4014, with our notation for the disjoint union of two sets (df-dju 7384). (Contributed by Jim Kingdon, 2-Oct-2026.)
⊢ 𝐹 = if(𝑥 = ∅, 𝐴, 𝐵)    ⇒   ⊢ ∪ 𝑥 ∈ 2o ({𝑥} × 𝐹) = (𝐴 ⊔ 𝐵)
14.4.6  Omniscience of NN+oo
Theorem	0nninf 17270	The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.)
⊢ (ω × {∅}) ∈ ℕ∞
Theorem	nnsf 17271*	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 17272*	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 17273*	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	nninfalllem1 17274*	Lemma for nninfall 17275. (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 17275*	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	nninfsellemdc 17276*	Lemma for nninfself 17279. 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 17277*	Lemma for nninfself 17279. (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 17278*	Lemma for nninfself 17279. (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 17279*	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 17280*	Lemma for nninfsel 17283. (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 17281*	Lemma for nninfsel 17283. (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 17282*	Lemma for nninfsel 17283. (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 17283*	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 17284*	Lemma for nninfomni 17285. (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 17285	ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
⊢ ℕ∞ ∈ Omni
Theorem	nninffeq 17286*	Equality of two functions on ℕ∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, ⊢ (𝜑 → ∀𝑛 ∈ suc ω...). (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	nnnninfen 17287	Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
⊢ (ω ≈ ℕ∞ ↔ ω ∈ Omni)
Theorem	nnnninfex 17288*	If an element of ℕ∞ has a value of zero somewhere, then it is the mapping of a natural number. (Contributed by Jim Kingdon, 4-Aug-2022.)
⊢ (𝜑 → 𝑃 ∈ ℕ∞)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → (𝑃‘𝑁) = ∅)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
Theorem	nninfnfiinf 17289*	An element of ℕ∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.)
⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → 𝐴 = (𝑖 ∈ ω ↦ 1o))
14.4.7  Schroeder-Bernstein Theorem
Theorem	exmidsbthrlem 17290*	Lemma for exmidsbthr 17291. (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 17291*	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 17292*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7291) 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-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 7291. The reverse direction (exmidsbthr 17291) 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 17293	Lemma for sbthom 17294. (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 17294	Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 17292 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)
14.4.8  Real and complex numbers
Theorem	qdencn 17295*	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 11990 (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 17296*	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	repiecelem 17297*	Lemma for repiecele0 17298, repiecege0 17299, and repiecef 17300. The function 𝐻 is defined everywhere. (Contributed by Jim Kingdon, 27-Apr-2026.)
⊢ (𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   ⊢ (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (((𝐹‘inf({𝐴, 0}, ℝ, < )) + (𝐺‘sup({𝐴, 0}, ℝ, < ))) − (𝐹‘0)) ∈ ℝ)
Theorem	repiecele0 17298*	Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 17300 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
⊢ (𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   ⊢ (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (𝐻‘𝐴) = (𝐹‘𝐴))
Theorem	repiecege0 17299*	Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 17300 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
⊢ (𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   ⊢ (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐻‘𝐴) = (𝐺‘𝐴))
Theorem	repiecef 17300*	Piecewise definition on the reals yields a function. The function agrees with 𝐹 and 𝐺 on their respective parts of the real line; see repiecele0 17298 and repiecege0 17299. From an online post by James E Hanson. The construction was published in Martín Hötzel Escardó, "Effective and sequential definition by cases on the reals via infinite signed-digit numerals", Electronic Notes in Theoretical Computer Science 10 (1998), page 2, https://martinescardo.github.io/papers/lexnew.pdf. 17299 (Contributed by Jim Kingdon, 27-Apr-2026.)
⊢ (𝜑 → 𝐹:(-∞(,]0)⟶ℝ)    &   ⊢ (𝜑 → 𝐺:(0[,)+∞)⟶ℝ)    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (((𝐹‘inf({𝑥, 0}, ℝ, < )) + (𝐺‘sup({𝑥, 0}, ℝ, < ))) − (𝐹‘0)))    ⇒   ⊢ (𝜑 → 𝐻:ℝ⟶ℝ)