P. 76 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	en2other2 7501	Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)
Theorem	dju1p1e2 7502	Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (1o ⊔ 1o) ≈ 2o
Theorem	infpwfidom 7503	The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
Theorem	exmidfodomrlemeldju 7504	Lemma for exmidfodomr 7509. A variant of djur 7362. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ (𝜑 → 𝐴 ⊆ 1o)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))    ⇒   ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
Theorem	exmidfodomrlemreseldju 7505	Lemma for exmidfodomrlemrALT 7508. A variant of eldju 7361. (Contributed by Jim Kingdon, 9-Jul-2022.)
⊢ (𝜑 → 𝐴 ⊆ 1o)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))    ⇒   ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
Theorem	exmidfodomrlemim 7506*	Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (EXMID → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))
Theorem	exmidfodomrlemr 7507*	The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦) → EXMID)
Theorem	exmidfodomrlemrALT 7508*	The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7507. In particular, this proof uses eldju 7361 instead of djur 7362 and avoids djulclb 7348. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
⊢ (∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦) → EXMID)
Theorem	exmidfodomr 7509*	Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (EXMID ↔ ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))
Theorem	acnrcl 7510	Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)
Theorem	acneq 7511	Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)
Theorem	isacnm 7512*	The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
Theorem	finacn 7513	Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 ∈ Fin → AC 𝐴 = V)
2.6.45  Axiom of Choice equivalents
Syntax	wac 7514	Formula for an abbreviation of the axiom of choice.
wff CHOICE
Definition	df-ac 7515*	The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49. There are some decisions about how to write this definition especially around whether ax-setind 4661 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.)
⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥))
Theorem	acfun 7516*	A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
⊢ (𝜑 → CHOICE)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
Theorem	exmidaclem 7517*	Lemma for exmidac 7518. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝑦 = {∅})}    &   ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝑦 = {∅})}    &   ⊢ 𝐶 = {𝐴, 𝐵}    ⇒   ⊢ (CHOICE → EXMID)
Theorem	exmidac 7518	The axiom of choice implies excluded middle. See acexmid 6051 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
⊢ (CHOICE → EXMID)
2.6.46  Cardinal number arithmetic
Theorem	endjudisj 7519	Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵))
Theorem	djuen 7520	Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))
Theorem	djuenun 7521	Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ∪ 𝐷))
Theorem	dju1en 7522	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴)
Theorem	dju0en 7523	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴)
Theorem	xp2dju 7524	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴)
Theorem	djucomen 7525	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))
Theorem	djuassen 7526	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶)))
Theorem	xpdjuen 7527	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))
Theorem	djudoml 7528	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵))
Theorem	djudomr 7529	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ≼ (𝐴 ⊔ 𝐵))
2.6.47  Ordinal trichotomy
Theorem	exmidontriimlem1 7530	Lemma for exmidontriim 7534. A variation of r19.30dc 2692. (Contributed by Jim Kingdon, 12-Aug-2024.)
⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	exmidontriimlem2 7531*	Lemma for exmidontriim 7534. (Contributed by Jim Kingdon, 12-Aug-2024.)
⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → EXMID)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑦 ∈ 𝐵 𝑦 ∈ 𝐴))
Theorem	exmidontriimlem3 7532*	Lemma for exmidontriim 7534. What we get to do based on induction on both 𝐴 and 𝐵. (Contributed by Jim Kingdon, 10-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → EXMID)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
Theorem	exmidontriimlem4 7533*	Lemma for exmidontriim 7534. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → EXMID)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
Theorem	exmidontriim 7534*	Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
⊢ (EXMID → ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
2.6.48  Excluded middle and the power set of a singleton
Theorem	iftrueb01 7535	Using an if expression to represent a truth value by ∅ or 1o. Unlike some theorems using if, 𝜑 does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)
⊢ (if(𝜑, 1o, ∅) = 1o ↔ 𝜑)
Theorem	pw1m 7536*	A truth value which is inhabited is equal to true. This is a variation of pwntru 4314 and pwtrufal 16820. (Contributed by Jim Kingdon, 10-Jan-2026.)
⊢ ((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)
Theorem	pw1if 7537	Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)
⊢ (𝐴 ∈ 𝒫 1o → if(𝐴 = 1o, 1o, ∅) = 𝐴)
Theorem	pw1on 7538	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
⊢ 𝒫 1o ∈ On
Theorem	pw1dom2 7539	The power set of 1o dominates 2o. Also see pwpw0ss 3911 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
⊢ 2o ≼ 𝒫 1o
Theorem	pw1ne0 7540	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
⊢ 𝒫 1o ≠ ∅
Theorem	pw1ne1 7541	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
⊢ 𝒫 1o ≠ 1o
Theorem	pw1ne3 7542	The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
⊢ 𝒫 1o ≠ 3o
Theorem	pw1nel3 7543	Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
⊢ (¬ EXMID → ¬ 𝒫 1o ∈ 3o)
Theorem	sucpw1ne3 7544	Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
⊢ (¬ EXMID → suc 𝒫 1o ≠ 3o)
Theorem	sucpw1nel3 7545	The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
⊢ ¬ suc 𝒫 1o ∈ 3o
Theorem	3nelsucpw1 7546	Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
⊢ ¬ 3o ∈ suc 𝒫 1o
Theorem	sucpw1nss3 7547	Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
⊢ (¬ EXMID → ¬ suc 𝒫 1o ⊆ 3o)
Theorem	3nsssucpw1 7548	Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
⊢ (¬ EXMID → ¬ 3o ⊆ suc 𝒫 1o)
Theorem	onntri35 7549*	Double negated ordinal trichotomy. There are five equivalent statements: (1) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (2) ¬ ¬ ∀𝑥 ∈ On∀𝑦 ∈ On(𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), (3) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥), (4) ∀𝑥 ∈ On∀𝑦 ∈ On¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥), and (5) ¬ ¬ EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7550), (3) implies (5) (onntri35 7549), (5) implies (1) (onntri51 7552), (2) implies (4) (onntri24 7554), (4) implies (5) (onntri45 7553), and (5) implies (2) (onntri52 7556). Another way of stating this is that EXMID is equivalent to trichotomy, either the 𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥 or the 𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥 form, as shown in exmidontri 7551 and exmidontri2or 7555, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7557 and (4) by onntri2or 7558. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)
Theorem	onntri13 7550	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
Theorem	exmidontri 7551*	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
Theorem	onntri51 7552*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
Theorem	onntri45 7553*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ¬ ¬ EXMID)
Theorem	onntri24 7554	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
Theorem	exmidontri2or 7555*	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
⊢ (EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
Theorem	onntri52 7556*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (¬ ¬ EXMID → ¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
Theorem	onntri3or 7557*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
Theorem	onntri2or 7558*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
Theorem	fmelpw1o 7559	With a formula 𝜑 one can associate an element of 𝒫 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than ⊤ and ⊥, by nndc 859, which translate to 1o and ∅ respectively by iftrue 3629 and iffalse 3632, giving pwtrufal 16820). As proved in if0ab 3625, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)
⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o
2.6.49  Apartness relations
Syntax	wap 7560	Apartness predicate symbol.
wff 𝑅 Ap 𝐴
Definition	df-pap 7561*	Apartness predicate. A relation 𝑅 is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.)
⊢ (𝑅 Ap 𝐴 ↔ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)))))
Theorem	papeq1 7562	Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
⊢ (𝑅 = 𝑆 → (𝑅 Ap 𝐴 ↔ 𝑆 Ap 𝐴))
Theorem	papeq2 7563	Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
⊢ (𝐴 = 𝐵 → (𝑅 Ap 𝐴 ↔ 𝑅 Ap 𝐵))
Theorem	papirr 7564	An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)
⊢ ((𝑅 Ap 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
Theorem	papsym 7565	An apartness is symmetric. (Contributed by Jim Kingdon, 27-May-2026.)
⊢ (𝜑 → 𝑅 Ap 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑅𝑌)    ⇒   ⊢ (𝜑 → 𝑌𝑅𝑋)
Theorem	papcotr 7566	An apartness is cotransitive. (Contributed by Jim Kingdon, 28-May-2026.)
⊢ (𝜑 → 𝑅 Ap 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑅𝑌)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋𝑅𝑍 ∨ 𝑌𝑅𝑍))
Syntax	wtap 7567	Tight apartness predicate symbol.
wff 𝑅 TAp 𝐴
Definition	df-tap 7568*	Tight apartness predicate. A relation 𝑅 is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.)
⊢ (𝑅 TAp 𝐴 ↔ (𝑅 Ap 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦)))
Theorem	tapap 7569	A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)
⊢ (𝑅 TAp 𝐴 → 𝑅 Ap 𝐴)
Theorem	dftap2 7570*	Tight apartness with the apartness properties from df-pap 7561 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
⊢ (𝑅 TAp 𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑦𝑅𝑧)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (¬ 𝑥𝑅𝑦 → 𝑥 = 𝑦))))
Theorem	tapeq1 7571	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
⊢ (𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
Theorem	tapeq2 7572	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
⊢ (𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵))
Theorem	netap 7573*	Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)} TAp 𝐴)
Theorem	2onetap 7574*	Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o
Theorem	2oneel 7575*	∅ and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
Theorem	2omotaplemap 7576*	Lemma for 2omotap 7578. (Contributed by Jim Kingdon, 6-Feb-2025.)
⊢ (¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
Theorem	2omotaplemst 7577*	Lemma for 2omotap 7578. (Contributed by Jim Kingdon, 6-Feb-2025.)
⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)
Theorem	2omotap 7578	If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)
Theorem	exmidapne 7579*	Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
⊢ (EXMID → (𝑅 TAp 𝐴 ↔ 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑢 ≠ 𝑣)}))
Theorem	exmidmotap 7580*	The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
⊢ (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
PART 3  CHOICE PRINCIPLES We have already introduced the full Axiom of Choice df-ac 7515 but since it implies excluded middle as shown at exmidac 7518, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle.
3.1  Countable Choice and Dependent Choice
3.1.1  Introduce Countable Choice
Syntax	wacc 7581	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7582*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7515 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
⊢ (CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)))
Theorem	ccfunen 7583*	Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐴 ≈ ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑤 𝑤 ∈ 𝑥)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
Theorem	cc1 7584*	Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
⊢ (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑓‘𝑧) ∈ 𝑧))
Theorem	cc2lem 7585*	Lemma for cc2 7586. (Contributed by Jim Kingdon, 27-Apr-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐹 Fn ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    &   ⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹‘𝑛)))    &   ⊢ 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴‘𝑛))))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
Theorem	cc2 7586*	Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐹 Fn ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔‘𝑛) ∈ (𝐹‘𝑛)))
Theorem	cc3 7587*	Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑁 ≈ ω)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))
Theorem	cc4f 7588*	Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ Ⅎ𝑛𝐴    &   ⊢ (𝜑 → 𝑁 ≈ ω)    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
Theorem	cc4 7589*	Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ≈ ω)    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜒))
Theorem	cc4n 7590*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7589, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
⊢ (𝜑 → CCHOICE)    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ≈ ω)    &   ⊢ (𝑥 = (𝑓‘𝑛) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 𝜒))
Theorem	acnccim 7591	Given countable choice, every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (CCHOICE → AC ω = V)
PART 4  REAL AND COMPLEX NUMBERS This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6709 and similar theorems ), going from there to positive integers (df-ni 7624) and then positive rational numbers (df-nqqs 7668) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle or choice principles. With excluded middle, it is natural to define a cut as the lower set only (as Metamath Proof Explorer does), but here we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". When working constructively, there are several possible definitions of real numbers. Here we adopt the most common definition, as two-sided Dedekind cuts with the properties described at df-inp 7786. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 8252 and the MacNeille reals fail to satisfy axltwlin 8346, and we do not develop them here. For more on differing definitions of the reals, see the introduction to Chapter 11 in [HoTT] or Section 1.2 of [BauerHanson].
4.1  Construction and axiomatization of real and complex numbers
4.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 7592	The set of positive integers, which is the set of natural numbers ω with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers.
class N
Syntax	cpli 7593	Positive integer addition.
class +N
Syntax	cmi 7594	Positive integer multiplication.
class ·N
Syntax	clti 7595	Positive integer ordering relation.
class <N
Syntax	cplpq 7596	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7597	Positive pre-fraction multiplication.
class ·pQ
Syntax	cltpq 7598	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7599	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7600	Set of positive fractions.
class Q