P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-acnm 7301*	Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋 ∈ AC 𝐴 is that for all families of inhabited subsets of 𝑋 indexed on 𝐴 (i.e. functions 𝐴⟶{𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗𝑗 ∈ 𝑧}), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) Change nonempty to inhabited. (Revised by Jim Kingdon, 22-Nov-2025.)
⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
Theorem	cardcl 7302*	The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)
Theorem	isnumi 7303	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)
Theorem	finnum 7304	Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
Theorem	onenon 7305	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
Theorem	cardval3ex 7306*	The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
Theorem	oncardval 7307*	The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
Theorem	cardonle 7308	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
Theorem	card0 7309	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
⊢ (card‘∅) = ∅
Theorem	ficardon 7310	The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ On)
Theorem	carden2bex 7311*	If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))
Theorem	pm54.43 7312	Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))
Theorem	pr2nelem 7313	Lemma for pr2ne 7314. (Contributed by FL, 17-Aug-2008.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
Theorem	pr2ne 7314	If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))
Theorem	pr1or2 7315	An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ DECID 𝐴 = 𝐵) → ({𝐴, 𝐵} ≈ 1o ∨ {𝐴, 𝐵} ≈ 2o))
Theorem	exmidonfinlem 7316*	Lemma for exmidonfin 7317. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   ⊢ (ω = (On ∩ Fin) → DECID 𝜑)
Theorem	exmidonfin 7317	If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6983 and nnon 4665. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
⊢ (ω = (On ∩ Fin) → EXMID)
Theorem	en2eleq 7318	Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})
Theorem	en2other2 7319	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 7320	Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (1o ⊔ 1o) ≈ 2o
Theorem	infpwfidom 7321	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 7322	Lemma for exmidfodomr 7327. A variant of djur 7185. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ (𝜑 → 𝐴 ⊆ 1o)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))    ⇒   ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
Theorem	exmidfodomrlemreseldju 7323	Lemma for exmidfodomrlemrALT 7326. A variant of eldju 7184. (Contributed by Jim Kingdon, 9-Jul-2022.)
⊢ (𝜑 → 𝐴 ⊆ 1o)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))    ⇒   ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
Theorem	exmidfodomrlemim 7324*	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 7325*	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 7326*	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 7325. In particular, this proof uses eldju 7184 instead of djur 7185 and avoids djulclb 7171. (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 7327*	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 7328	Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)
Theorem	acneq 7329	Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)
Theorem	isacnm 7330*	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 7331	Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 ∈ Fin → AC 𝐴 = V)
2.6.42  Axiom of Choice equivalents
Syntax	wac 7332	Formula for an abbreviation of the axiom of choice.
wff CHOICE
Definition	df-ac 7333*	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 4592 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 7334*	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 7335*	Lemma for exmidac 7336. 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 7336	The axiom of choice implies excluded middle. See acexmid 5955 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)
2.6.43  Cardinal number arithmetic
Theorem	endjudisj 7337	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 7338	Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))
Theorem	djuenun 7339	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 7340	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 7341	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 7342	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 7343	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 7344	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 7345	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 7346	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 7347	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ≼ (𝐴 ⊔ 𝐵))
2.6.44  Ordinal trichotomy
Theorem	exmidontriimlem1 7348	Lemma for exmidontriim 7352. A variation of r19.30dc 2654. (Contributed by Jim Kingdon, 12-Aug-2024.)
⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ∧ EXMID) → (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	exmidontriimlem2 7349*	Lemma for exmidontriim 7352. (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 7350*	Lemma for exmidontriim 7352. 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 7351*	Lemma for exmidontriim 7352. 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 7352*	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))
2.6.45  Excluded middle and the power set of a singleton
Theorem	pw1on 7353	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
⊢ 𝒫 1o ∈ On
Theorem	pw1dom2 7354	The power set of 1o dominates 2o. Also see pwpw0ss 3850 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
⊢ 2o ≼ 𝒫 1o
Theorem	pw1ne0 7355	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
⊢ 𝒫 1o ≠ ∅
Theorem	pw1ne1 7356	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
⊢ 𝒫 1o ≠ 1o
Theorem	pw1ne3 7357	The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
⊢ 𝒫 1o ≠ 3o
Theorem	pw1nel3 7358	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 7359	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 7360	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 7361	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 7362	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 7363	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 7364*	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 7365), (3) implies (5) (onntri35 7364), (5) implies (1) (onntri51 7367), (2) implies (4) (onntri24 7369), (4) implies (5) (onntri45 7368), and (5) implies (2) (onntri52 7371). 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 7366 and exmidontri2or 7370, respectively. Thus ¬ ¬ EXMID is equivalent to (1) or (2). In addition, ¬ ¬ EXMID is equivalent to (3) by onntri3or 7372 and (4) by onntri2or 7373. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ¬ ¬ EXMID)
Theorem	onntri13 7365	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 7366*	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 7367*	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 7368*	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 7369	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 7370*	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 7371*	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 7372*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
Theorem	onntri2or 7373*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
⊢ (¬ ¬ EXMID ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
2.6.46  Apartness relations
Syntax	wap 7374	Apartness predicate symbol.
wff 𝑅 Ap 𝐴
Definition	df-pap 7375*	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)))))
Syntax	wtap 7376	Tight apartness predicate symbol.
wff 𝑅 TAp 𝐴
Definition	df-tap 7377*	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	dftap2 7378*	Tight apartness with the apartness properties from df-pap 7375 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 7379	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
⊢ (𝑅 = 𝑆 → (𝑅 TAp 𝐴 ↔ 𝑆 TAp 𝐴))
Theorem	tapeq2 7380	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
⊢ (𝐴 = 𝐵 → (𝑅 TAp 𝐴 ↔ 𝑅 TAp 𝐵))
Theorem	netap 7381*	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 7382*	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 7383*	∅ 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 7384*	Lemma for 2omotap 7386. (Contributed by Jim Kingdon, 6-Feb-2025.)
⊢ (¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
Theorem	2omotaplemst 7385*	Lemma for 2omotap 7386. (Contributed by Jim Kingdon, 6-Feb-2025.)
⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)
Theorem	2omotap 7386	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 7387*	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 7388*	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 7333 but since it implies excluded middle as shown at exmidac 7336, 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 7389	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7390*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7333 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
⊢ (CCHOICE ↔ ∀𝑥(dom 𝑥 ≈ ω → ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)))
Theorem	ccfunen 7391*	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 7392*	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 7393*	Lemma for cc2 7394. (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 7394*	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 7395*	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 7396*	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 7397*	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 7398*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7397, 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 7399	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 6572 and similar theorems ), going from there to positive integers (df-ni 7432) and then positive rational numbers (df-nqqs 7476) 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 7594. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 8060 and the MacNeille reals fail to satisfy axltwlin 8155, 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 7400	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