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
2.4.29  Equinumerosity
Syntax	cen 9001	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ≈
Syntax	cdom 9002	Extend class definition to include the dominance relation (curly "less than or equal to")
class ≼
Syntax	csdm 9003	Extend class definition to include the strict dominance relation (curly less-than)
class ≺
Syntax	cfn 9004	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 9005*	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 9014. (Contributed by NM, 28-Mar-1998.)
⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1onto𝑦}
Definition	df-dom 9006*	Define the dominance relation. For an alternate definition see dfdom2 9036. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 9018 and domen 9019. (Contributed by NM, 28-Mar-1998.)
⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥⟶1-1𝑦}
Definition	df-sdom 9007	Define the strict dominance relation. Alternate possible definitions are derived as brsdom 9032 and brsdom2 9151. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
⊢ ≺ = ( ≼ ∖ ≈ )
Definition	df-fin 9008*	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 9672. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (Theorem isfinite 9683.) (Contributed by NM, 22-Aug-2008.)
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
Theorem	relen 9009	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≈
Theorem	reldom 9010	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
⊢ Rel ≼
Theorem	relsdom 9011	Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ Rel ≺
Theorem	encv 9012	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	breng 9013*	Equinumerosity relation. This variation of bren 9014 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 9014. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
Theorem	bren 9014*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) Extract breng 9013 as an intermediate result. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
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⟶1-1𝐵))
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⟶1-1𝐵))
Theorem	brdomi 9017*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7778. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴⟶1-1𝐵)
Theorem	brdom 9018*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵)
Theorem	domen 9019*	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 9020*	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 9021	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 9022	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9028 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⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1dom4g 9023	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 9029 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⟶1-1𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oen3g 9024	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 9028 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⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1dom3g 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 9029 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⟶1-1𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oen2g 9026	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 9028 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1dom2g 9027	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 9029 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⟶1-1𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1oeng 9028	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1domg 9029	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶1-1𝐵 → 𝐴 ≼ 𝐵))
Theorem	f1oen 9030	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐴 ≈ 𝐵)
Theorem	f1dom 9031	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐴 ≼ 𝐵)
Theorem	brsdom 9032	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 9033*	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 9034	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) (Proof shortened by TM, 10-Feb-2026.)
⊢ ≈ ⊆ ≼
Theorem	enssdomOLD 9035	Obsolete version of enssdom 9034 as of 10-Feb-2026. (Contributed by NM, 31-Mar-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ≈ ⊆ ≼
Theorem	dfdom2 9036	Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
⊢ ≼ = ( ≺ ∪ ≈ )
Theorem	endom 9037	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵)
Theorem	sdomdom 9038	Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
Theorem	sdomnen 9039	Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
Theorem	brdom2 9040	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 9041	Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵))
Theorem	enrefg 9042	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 9043	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ≈ 𝐴
Theorem	eqeng 9044	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
Theorem	domrefg 9045	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴)
Theorem	en2d 9046*	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 9047*	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 9048*	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 9049*	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 9050*	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⟶1-1𝐵)
Theorem	dom2d 9051*	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 9052*	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 9053*	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 9054*	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 9055	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ I ⊆ ≈
Theorem	domssl 9056	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 9057	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 9058	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 9059	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 9060	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)
Theorem	ensym 9061	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 9062	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    ⇒   ⊢ 𝐵 ≈ 𝐴
Theorem	ensymd 9063	Symmetry of equinumerosity. Deduction form of ensym 9061. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≈ 𝐴)
Theorem	entr 9064	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	domtr 9065	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 9066	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐵 ≈ 𝐶    ⇒   ⊢ 𝐴 ≈ 𝐶
Theorem	entr2i 9067	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐵 ≈ 𝐶    ⇒   ⊢ 𝐶 ≈ 𝐴
Theorem	entr3i 9068	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐴 ≈ 𝐶    ⇒   ⊢ 𝐵 ≈ 𝐶
Theorem	entr4i 9069	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
⊢ 𝐴 ≈ 𝐵    &   ⊢ 𝐶 ≈ 𝐵    ⇒   ⊢ 𝐴 ≈ 𝐶
Theorem	endomtr 9070	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domentr 9071	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	f1imaeng 9072	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⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	f1imaen2g 9073	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 9072 does not need ax-rep 5236.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ (((𝐹:𝐴⟶1-1𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	f1imaen3g 9074	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 9072 does not need ax-rep 5236 nor ax-pow 5332.) (Contributed by BTernaryTau, 13-Jan-2025.)
⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐹 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))
Theorem	f1imaen 9075	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⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	en0 9076	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5332, ax-un 7778. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Theorem	en0ALT 9077	Shorter proof of en0 9076, depending on ax-pow 5332 and ax-un 7778. (Contributed by NM, 27-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Theorem	en0r 9078	The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)
⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)
Theorem	ensn1 9079	A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7778. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≈ 1o
Theorem	ensn1g 9080	A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)
Theorem	enpr1g 9081	{𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o)
Theorem	en1 9082*	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7778. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	en1b 9083	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7778. (Revised by BTernaryTau, 24-Sep-2024.)
⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})
Theorem	reuen1 9084*	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o)
Theorem	euen1 9085	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o)
Theorem	euen1b 9086*	Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐴 ≈ 1o ↔ ∃!𝑥 𝑥 ∈ 𝐴)
Theorem	funen1cnv 9087	If a function is equinumerous to ordinal 1, then its converse is also a function. (Contributed by BTernaryTau, 8-Oct-2023.)
⊢ ((Fun 𝐹 ∧ 𝐹 ≈ 1o) → Fun ◡𝐹)
Theorem	en1uniel 9088	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7778. (Revised by BTernaryTau, 24-Sep-2024.)
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
Theorem	2dom 9089*	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 9090	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 9091	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 9092	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴)
Theorem	cnvct 9093	If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω)
Theorem	fndmeng 9094	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹)
Theorem	mapsnend 9095	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.) (Revised by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴)
Theorem	mapsnen 9096	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.) (Proof shortened by AV, 17-Jul-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ↑m {𝐵}) ≈ 𝐴
Theorem	snmapen 9097	Set exponentiation: a singleton to any set is equinumerous to that singleton. (Contributed by NM, 17-Dec-2003.) (Revised by AV, 17-Jul-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})
Theorem	snmapen1 9098	Set exponentiation: a singleton to any set is equinumerous to ordinal 1. (Proposed by BJ, 17-Jul-2022.) (Contributed by AV, 17-Jul-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ 1o)
Theorem	map1 9099	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.) (Proof shortened by AV, 17-Jul-2022.)
⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o)
Theorem	en2sn 9100	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5332. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7778. (Revised by BTernaryTau, 25-Sep-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})