P. 172 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 17101-17200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-omtrans 17101	The set ω is transitive. A natural number is included in ω. Constructive proof of elomssom 4752. 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 17102	The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Tr ω
Theorem	bj-nnord 17103	A natural number is an ordinal class. Constructive proof of nnord 4759. Can also be proved from bj-nnelon 17104 if the latter is proved from bj-omssonALT 17108. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	bj-nnelon 17104	A natural number is an ordinal. Constructive proof of nnon 4757. Can also be proved from bj-omssonALT 17108. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	bj-omord 17105	The set ω is an ordinal class. Constructive proof of ordom 4754. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ Ord ω
Theorem	bj-omelon 17106	The set ω is an ordinal. Constructive proof of omelon 4756. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
⊢ ω ∈ On
Theorem	bj-omsson 17107	Constructive proof of omsson 4760. See also bj-omssonALT 17108. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
⊢ ω ⊆ On
Theorem	bj-omssonALT 17108	Alternate proof of bj-omsson 17107. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ⊆ On
Theorem	bj-nn0suc 17109*	Proof of (biconditional form of) nn0suc 4751 from the core axioms of CZF. See also bj-nn0sucALT 17123. 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 17110*	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 17111*	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 17112*	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 17113*	Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
Theorem	bdsetindis 17114*	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 17115*	Lemma for bj-inf2vn 17119. 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 17116*	Lemma for bj-inf2vnlem3 17117 and bj-inf2vnlem4 17118. 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 17117*	Lemma for bj-inf2vn 17119. (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 17118*	Lemma for bj-inf2vn2 17120. (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 17119*	A sufficient condition for ω to be a set. See bj-inf2vn2 17120 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 17120*	A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 17119. (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 17121*	Another axiom of infinity in a constructive setting (see ax-infvn 17086). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.)
⊢ ∃𝑎∀𝑥(𝑥 ∈ 𝑎 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝑎 𝑥 = suc 𝑦))
Theorem	bj-omex2 17122	Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 17086 (see bj-2inf 17083 for the equivalence of the latter with bj-omex 17087). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ω ∈ V
Theorem	bj-nn0sucALT 17123*	Alternate proof of bj-nn0suc 17109, also constructive but from ax-inf2 17121, hence requiring ax-bdsetind 17113. (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 17124*	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 17092 for a bounded version not requiring ax-setind 4684. See finds 4747 for a proof in IZF. From this version, it is easy to prove of finds 4747, finds2 4748, finds1 4749. (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 17125*	Version of bj-findis 17124 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 17124 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 17126	Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 17124 for explanations. From this version, it is easy to prove findes 4750. (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 17127*	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 4246. (Contributed by BJ, 5-Oct-2019.)
⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
Theorem	strcoll2 17128*	Version of ax-strcoll 17127 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 17129*	Closed form of strcollnf 17130. (Contributed by BJ, 21-Oct-2019.)
⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
Theorem	strcollnf 17130*	Version of ax-strcoll 17127 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 17128 with the disjoint variable condition on 𝑏, 𝜑 replaced with a nonfreeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 17128 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 17131*	Alternate proof of strcollnf 17130, not using strcollnft 17129. (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 17132*	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 17133*	Version of ax-sscoll 17132 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 17134	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 17134 should be used in place of construction specific results. In particular, axcaucvg 8267 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 17135	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 17136	Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4335 which more directly illustrates the contrast with el2oss1o 6716. (Contributed by Jim Kingdon, 8-Aug-2022.)
⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o))
Theorem	3dom 17137*	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 17138	Lemma for pw1ndom3 17139. (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 17139	The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4336). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
⊢ ¬ 3o ≼ 𝒫 1o
Theorem	pw1ninf 17140	The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7217), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7213. (Contributed by Jim Kingdon, 14-Feb-2026.)
⊢ ¬ ω ≼ 𝒫 1o
Theorem	nnti 17141	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 17142	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 17143	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 17144*	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 17145	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 17146	A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4335. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4333), 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 17147*	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 17148*	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 17149*	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 17150*	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 17151*	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 17152*	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 17153*	Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)
⊢ ({𝑥 ∈ 1o ∣ 𝜑} = 1o ↔ 𝜑)
Theorem	pw1dceq 17154*	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 17155	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 17156*	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 17157*	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 17158*	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 17159	Formula for an abbreviation of weak excluded middle.
wff WEXMID
Definition	df-wexmid 17160	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 17161	Weak excluded middle expressed using WEXMID implies decidability of a negated proposition. (Contributed by Jim Kingdon, 30-Jul-2026.)
⊢ (WEXMID → DECID ¬ 𝜑)
Theorem	wexmiddiffilem 17162*	Lemma for wexmiddiffi 17163. The reverse direction, using different notation. (Contributed by Jim Kingdon, 29-Jul-2026.)
⊢ (∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
Theorem	wexmiddiffi 17163*	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 7198. (Contributed by Jim Kingdon, 29-Jul-2026.)
⊢ (WEXMID ↔ ∀𝑥∀𝑦(𝑥 ∈ Fin → (𝑥 ∖ 𝑦) ∈ Fin))
Theorem	wexmiddifxylem 17164*	Lemma for wexmiddifxylem 17164. 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 17165*	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 7198. (Contributed by Jim Kingdon, 1-Aug-2026.)
⊢ (WEXMID ↔ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∖ 𝑦) ∈ Fin))
14.4.5  Omniscience of NN+oo
Theorem	0nninf 17166	The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.)
⊢ (ω × {∅}) ∈ ℕ∞
Theorem	nnsf 17167*	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 17168*	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 17169*	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 17170*	Lemma for nninfall 17171. (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 17171*	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 17172*	Lemma for nninfself 17175. 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 17173*	Lemma for nninfself 17175. (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 17174*	Lemma for nninfself 17175. (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 17175*	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 17176*	Lemma for nninfsel 17179. (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 17177*	Lemma for nninfsel 17179. (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 17178*	Lemma for nninfsel 17179. (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 17179*	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 17180*	Lemma for nninfomni 17181. (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 17181	ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.)
⊢ ℕ∞ ∈ Omni
Theorem	nninffeq 17182*	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 17183	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 17184*	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 17185*	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))
14.4.6  Schroeder-Bernstein Theorem
Theorem	exmidsbthrlem 17186*	Lemma for exmidsbthr 17187. (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 17187*	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 17188*	The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7284) 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 7284. The reverse direction (exmidsbthr 17187) 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 17189	Lemma for sbthom 17190. (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 17190	Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 17188 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)
14.4.7  Real and complex numbers
Theorem	qdencn 17191*	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 11983 (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 17192*	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 17193*	Lemma for repiecele0 17194, repiecege0 17195, and repiecef 17196. 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 17194*	Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 17196 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 17195*	Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 17196 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 17196*	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 17194 and repiecege0 17195. 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. 17195 (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)
Theorem	triap 17197	Two ways of stating real number trichotomy. See also cndcap 17228 which is similar but for complex number apartness. (Contributed by Jim Kingdon, 23-Aug-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ DECID 𝐴 # 𝐵))
Theorem	isomninnlem 17198*	Lemma for isomninn 17199. 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)))
Theorem	isomninn 17199*	Omniscience stated in terms of natural numbers. Similar to isomnimap 7477 but it will sometimes be more convenient to use 0 and 1 rather than ∅ and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)))
Theorem	cvgcmp2nlemabs 17200*	Lemma for cvgcmp2n 17201. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting (seq1( + , 𝐺)‘𝑁) as the sum of (seq1( + , 𝐺)‘𝑀) and a term which gets smaller as 𝑀 gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))