P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	resixp 6801*	Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶)
Theorem	mptelixpg 6802*	Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
Theorem	elixpsn 6803*	Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
Theorem	ixpsnf1o 6804*	A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)
Theorem	mapsnf1o 6805*	A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑𝑚 {𝐼}))
2.6.28  Equinumerosity
Syntax	cen 6806	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ≈
Syntax	cdom 6807	Extend class definition to include the dominance relation (curly less-than-or-equal)
class ≼
Syntax	cfn 6808	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 6809*	Define the equinumerosity relation. Definition of [Enderton] p. 129. We define ≈ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 6815. (Contributed by NM, 28-Mar-1998.)
⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
Definition	df-dom 6810*	Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6818 and domen 6819. (Contributed by NM, 28-Mar-1998.)
⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
Definition	df-fin 6811*	Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our "𝑎 ∈ Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 15706. (Contributed by NM, 22-Aug-2008.)
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
Theorem	relen 6812	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≈
Theorem	reldom 6813	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≼
Theorem	encv 6814	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bren 6815*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
Theorem	brdomg 6816*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomi 6817*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
Theorem	brdom 6818*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)
Theorem	domen 6819*	Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))
Theorem	domeng 6820*	Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)))
Theorem	ctex 6821	A class dominated by ω is a set. See also ctfoex 7193 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
Theorem	f1oen3g 6822	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6825 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1oen2g 6823	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6825 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1dom2g 6824	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6826 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oeng 6825	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1domg 6826	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵))
Theorem	f1oen 6827	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵)
Theorem	f1dom 6828	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)
Theorem	isfi 6829*	Express "𝐴 is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
Theorem	enssdom 6830	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
⊢ ≈ ⊆ ≼
Theorem	endom 6831	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵)
Theorem	enrefg 6832	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
Theorem	enref 6833	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ≈ 𝐴
Theorem	eqeng 6834	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
Theorem	domrefg 6835	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴)
Theorem	en2d 6836*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V))    &   ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))    ⇒   ⊢ (𝜑 → 𝐴 ≈ 𝐵)
Theorem	en3d 6837*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))    ⇒   ⊢ (𝜑 → 𝐴 ≈ 𝐵)
Theorem	en2i 6838*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ V)    &   ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))    ⇒   ⊢ 𝐴 ≈ 𝐵
Theorem	en3i 6839*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)    &   ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))    ⇒   ⊢ 𝐴 ≈ 𝐵
Theorem	dom2lem 6840*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)
Theorem	dom2d 6841*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵))
Theorem	dom3d 6842*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐴 ≼ 𝐵)
Theorem	dom2 6843*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵)
Theorem	dom3 6844*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ 𝐵)
Theorem	idssen 6845	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ I ⊆ ≈
Theorem	ssdomg 6846	A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵))
Theorem	ener 6847	Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ ≈ Er V
Theorem	ensymb 6848	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)
Theorem	ensym 6849	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
Theorem	ensymi 6850	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    ⇒   ⊢ 𝐵 ≈ 𝐴
Theorem	ensymd 6851	Symmetry of equinumerosity. Deduction form of ensym 6849. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≈ 𝐴)
Theorem	entr 6852	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	domtr 6853	Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	entri 6854	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐵 ≈ 𝐶    ⇒   ⊢ 𝐴 ≈ 𝐶
Theorem	entr2i 6855	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐵 ≈ 𝐶    ⇒   ⊢ 𝐶 ≈ 𝐴
Theorem	entr3i 6856	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐴 ≈ 𝐶    ⇒   ⊢ 𝐵 ≈ 𝐶
Theorem	entr4i 6857	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐶 ≈ 𝐵    ⇒   ⊢ 𝐴 ≈ 𝐶
Theorem	endomtr 6858	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domentr 6859	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	f1imaeng 6860	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	f1imaen2g 6861	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6862 does not need ax-setind 4574.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	f1imaen 6862	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	en0 6863	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Theorem	ensn1 6864	A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≈ 1o
Theorem	ensn1g 6865	A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)
Theorem	enpr1g 6866	{𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o)
Theorem	en1 6867*	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	en1bg 6868	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}))
Theorem	reuen1 6869*	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o)
Theorem	euen1 6870	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o)
Theorem	euen1b 6871*	Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐴 ≈ 1o ↔ ∃!𝑥 𝑥 ∈ 𝐴)
Theorem	en1uniel 6872	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
Theorem	2dom 6873*	A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
Theorem	fundmen 6874	A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)
Theorem	fundmeng 6875	A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹)
Theorem	cnven 6876	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴)
Theorem	cnvct 6877	If a set is dominated by ω, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω)
Theorem	fndmeng 6878	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)
Theorem	mapsnen 6879	Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴
Theorem	map1 6880	Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)
⊢ (𝐴 ∈ 𝑉 → (1o ↑𝑚 𝐴) ≈ 1o)
Theorem	en2sn 6881	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})
Theorem	snfig 6882	A singleton is finite. For the proper class case, see snprc 3688. (Contributed by Jim Kingdon, 13-Apr-2020.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin)
Theorem	fiprc 6883	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
⊢ Fin ∉ V
Theorem	unen 6884	Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
Theorem	enpr2d 6885	A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o)
Theorem	ssct 6886	A subset of a set dominated by ω is dominated by ω. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)
Theorem	1domsn 6887	A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
⊢ {𝐴} ≼ 1o
Theorem	enm 6888*	A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)
Theorem	xpsnen 6889	A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × {𝐵}) ≈ 𝐴
Theorem	xpsneng 6890	A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)
Theorem	xp1en 6891	One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 × 1o) ≈ 𝐴)
Theorem	endisj 6892*	Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)
Theorem	xpcomf1o 6893*	The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})    ⇒   ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Theorem	xpcomco 6894*	Composition with the bijection of xpcomf1o 6893 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})    &   ⊢ 𝐺 = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶)    ⇒   ⊢ (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
Theorem	xpcomen 6895	Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
Theorem	xpcomeng 6896	Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
Theorem	xpsnen2g 6897	A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × 𝐵) ≈ 𝐵)
Theorem	xpassen 6898	Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶))
Theorem	xpdom2 6899	Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
Theorem	xpdom2g 6900	Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≼ 𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))