P. 91 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cdom 9001	Extend class definition to include the dominance relation (curly "less than or equal to")
class ≼
Syntax	csdm 9002	Extend class definition to include the strict dominance relation (curly less-than)
class ≺
Syntax	cfn 9003	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 9004*	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 9013. (Contributed by NM, 28-Mar-1998.)
⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
Definition	df-dom 9005*	Define the dominance relation. For an alternate definition see dfdom2 9038. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 9020 and domen 9021. (Contributed by NM, 28-Mar-1998.)
⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
Definition	df-sdom 9006	Define the strict dominance relation. Alternate possible definitions are derived as brsdom 9035 and brsdom2 9163. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
⊢ ≺ = ( ≼ ∖ ≈ )
Definition	df-fin 9007*	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 9710. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (Theorem isfinite 9721.) (Contributed by NM, 22-Aug-2008.)
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
Theorem	relen 9008	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≈
Theorem	reldom 9009	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≼
Theorem	relsdom 9010	Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ Rel ≺
Theorem	encv 9011	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	breng 9012*	Equinumerosity relation. This variation of bren 9013 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 9013. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
Theorem	bren 9013*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) Extract breng 9012 as an intermediate result. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
Theorem	brenOLD 9014*	Obsolete version of bren 9013 as of 23-Sep-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
Theorem	brdom2g 9015*	Dominance relation. This variation of brdomg 9016 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 9016. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomg 9016*	Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract brdom2g 9015 as an intermediate result. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomgOLD 9017*	Obsolete version of brdomg 9016 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
Theorem	brdomi 9018*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7770. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
Theorem	brdomiOLD 9019*	Obsolete version of brdomi 9018 as of 29-Nov-2024. (Contributed by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
Theorem	brdom 9020*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)
Theorem	domen 9021*	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 9022*	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 9023	A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
Theorem	f1oen4g 9024	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9031 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1dom4g 9025	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 9032 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oen3g 9026	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9031 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1dom3g 9027	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 9032 does not require the Axiom of Replacement nor the Axiom of Power Sets. (Contributed by BTernaryTau, 9-Sep-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oen2g 9028	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 9031 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 9029	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 9032 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof shortened by BTernaryTau, 25-Sep-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1dom2gOLD 9030	Obsolete version of f1dom2g 9029 as of 25-Sep-2024. (Contributed by Mario Carneiro, 24-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oeng 9031	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 9032	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 9033	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 9034	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	brsdom 9035	Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴". Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)
⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))
Theorem	isfi 9036*	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 9037	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
⊢ ≈ ⊆ ≼
Theorem	dfdom2 9038	Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
⊢ ≼ = ( ≺ ∪ ≈ )
Theorem	endom 9039	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵)
Theorem	sdomdom 9040	Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
Theorem	sdomnen 9041	Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
Theorem	brdom2 9042	Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))
Theorem	bren2 9043	Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵))
Theorem	enrefg 9044	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 9045	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ≈ 𝐴
Theorem	eqeng 9046	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
Theorem	domrefg 9047	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴)
Theorem	en2d 9048*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋))    &   ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))    ⇒   ⊢ (𝜑 → 𝐴 ≈ 𝐵)
Theorem	en3d 9049*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))    ⇒   ⊢ (𝜑 → 𝐴 ≈ 𝐵)
Theorem	en2i 9050*	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 9051*	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 9052*	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 9053*	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 9054*	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 9055*	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 9056*	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 9057	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ I ⊆ ≈
Theorem	domssl 9058	If 𝐴 is a subset of 𝐵 and 𝐶 dominates 𝐵, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domssr 9059	If 𝐶 is a superset of 𝐵 and 𝐵 dominates 𝐴, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶)
Theorem	ssdomg 9060	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 9061	Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 1-May-2021.)
⊢ ≈ Er V
Theorem	ensymb 9062	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)
Theorem	ensym 9063	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 9064	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    ⇒   ⊢ 𝐵 ≈ 𝐴
Theorem	ensymd 9065	Symmetry of equinumerosity. Deduction form of ensym 9063. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≈ 𝐴)
Theorem	entr 9066	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	domtr 9067	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 9068	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐵 ≈ 𝐶    ⇒   ⊢ 𝐴 ≈ 𝐶
Theorem	entr2i 9069	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐵 ≈ 𝐶    ⇒   ⊢ 𝐶 ≈ 𝐴
Theorem	entr3i 9070	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐴 ≈ 𝐶    ⇒   ⊢ 𝐵 ≈ 𝐶
Theorem	entr4i 9071	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐶 ≈ 𝐵    ⇒   ⊢ 𝐴 ≈ 𝐶
Theorem	endomtr 9072	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domentr 9073	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	f1imaeng 9074	If a function is one-to-one, then the image of a subset of its domain under it 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 9075	If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (This version of f1imaeng 9074 does not need ax-rep 5303.) (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	f1imaen3g 9076	If a set function is one-to-one, then a subset of its domain is equinumerous to the image of that subset. (This version of f1imaeng 9074 does not need ax-rep 5303 nor ax-pow 5383.) (Contributed by BTernaryTau, 13-Jan-2025.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))
Theorem	f1imaen 9077	If a function is one-to-one, then the image of a subset of its domain under it 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 9078	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5383, ax-un 7770. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Theorem	en0OLD 9079	Obsolete version of en0 9078 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5383. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Theorem	en0ALT 9080	Shorter proof of en0 9078, depending on ax-pow 5383 and ax-un 7770. (Contributed by NM, 27-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Theorem	en0r 9081	The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)
⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
Theorem	ensn1 9082	A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7770. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≈ 1o
Theorem	ensn1OLD 9083	Obsolete version of ensn1 9082 as of 23-Sep-2024. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≈ 1o
Theorem	ensn1g 9084	A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)
Theorem	enpr1g 9085	{𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o)
Theorem	en1 9086*	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7770. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	en1OLD 9087*	Obsolete version of en1 9086 as of 23-Sep-2024. (Contributed by NM, 25-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	en1b 9088	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7770. (Revised by BTernaryTau, 24-Sep-2024.)
⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})
Theorem	en1bOLD 9089	Obsolete version of en1b 9088 as of 24-Sep-2024. (Contributed by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})
Theorem	reuen1 9090*	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o)
Theorem	euen1 9091	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o)
Theorem	euen1b 9092*	Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐴 ≈ 1o ↔ ∃!𝑥 𝑥 ∈ 𝐴)
Theorem	en1uniel 9093	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7770. (Revised by BTernaryTau, 24-Sep-2024.)
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
Theorem	en1unielOLD 9094	Obsolete version of en1uniel 9093 as of 24-Sep-2024. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
Theorem	2dom 9095*	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 9096	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 9097	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 9098	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴)
Theorem	cnvct 9099	If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω)
Theorem	fndmeng 9100	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)