P. 43 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nelpri 4201	If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
Theorem	prneli 4202	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐴 ∉ {𝐵, 𝐶}
Theorem	nelprd 4203	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Theorem	eldifpr 4204	Membership in a set with two elements removed. Similar to eldifsn 4317 and eldiftp 4228. (Contributed by Mario Carneiro, 18-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
Theorem	rexdifpr 4205	Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))
Theorem	snidg 4206	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
Theorem	snidb 4207	A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Theorem	snid 4208	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ {𝐴}
Theorem	vsnid 4209	A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 𝑥 ∈ {𝑥}
Theorem	elsn2g 4210	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	elsn2 4211	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
Theorem	nelsn 4212	If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})
Theorem	nelsnOLD 4213	Obsolete proof of nelsn 4212 as of 4-May-2021. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})
Theorem	rabeqsn 4214*	Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))
Theorem	rabsssn 4215*	Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋))
Theorem	ralsnsg 4216*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	rexsns 4217*	Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	ralsng 4218*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	rexsng 4219*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	2ralsng 4220*	Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒))
Theorem	exsnrex 4221	There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥})
Theorem	ralsn 4222*	Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)
Theorem	rexsn 4223*	Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)
Theorem	elpwunsn 4224	Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)
Theorem	eqoreldif 4225	An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
Theorem	eqoreldifOLD 4226	Obsolete proof of eqoreldif 4225 as of 23-Jul-2021. (Contributed by AV, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
Theorem	eltpg 4227	Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
Theorem	eldiftp 4228	Membership in a set with three elements removed. Similar to eldifsn 4317 and eldifpr 4204. (Contributed by David A. Wheeler, 22-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
Theorem	eltpi 4229	A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
Theorem	eltp 4230	A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
Theorem	dftp2 4231*	Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
Theorem	nfpr 4232	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥{𝐴, 𝐵}
Theorem	ifpr 4233	Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
Theorem	ralprg 4234*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	rexprg 4235*	Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
Theorem	raltpg 4236*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	rextpg 4237*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃)))
Theorem	ralpr 4238*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))
Theorem	rexpr 4239*	Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))
Theorem	raltp 4240*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	rextp 4241*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃))
Theorem	nfsn 4242	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝐴}
Theorem	csbsng 4243	Distribute proper substitution through the singleton of a class. csbsng 4243 is derived from the virtual deduction proof csbsngVD 39129. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})
Theorem	csbprg 4244	Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
⊢ (𝐶 ∈ 𝑉 → ⦋𝐶 / 𝑥⦌{𝐴, 𝐵} = {⦋𝐶 / 𝑥⦌𝐴, ⦋𝐶 / 𝑥⦌𝐵})
Theorem	elinsn 4245	If the intersection of two classes is a (proper) singleton, then the singleton element is a member of both classes. (Contributed by AV, 30-Dec-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
Theorem	disjsn 4246	Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
Theorem	disjsn2 4247	Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998.)
⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Theorem	disjpr2 4248	Two completely distinct unordered pairs are disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Proof shortened by JJ, 23-Jul-2021.)
⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
Theorem	disjpr2OLD 4249	Obsolete proof of disjpr2 4248 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
Theorem	disjprsn 4250	The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
Theorem	disjtpsn 4251	The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.)
⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
Theorem	disjtp2 4252	Two completely distinct unordered triples are disjoint. (Contributed by AV, 14-Nov-2021.)
⊢ (((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐵 ≠ 𝐸 ∧ 𝐶 ≠ 𝐸) ∧ (𝐴 ≠ 𝐹 ∧ 𝐵 ≠ 𝐹 ∧ 𝐶 ≠ 𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
Theorem	snprc 4253	The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 21-Jun-1993.)
⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
Theorem	snnzb 4254	A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
Theorem	r19.12sn 4255*	Special case of r19.12 3063 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑))
Theorem	rabsn 4256*	Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})
Theorem	rabsnifsb 4257*	A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
Theorem	rabsnif 4258*	A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Theorem	rabrsn 4259*	A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)
⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
Theorem	euabsn2 4260*	Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
Theorem	euabsn 4261	Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥})
Theorem	reusn 4262*	A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})
Theorem	absneu 4263	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑)
Theorem	rabsneu 4264	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑)
Theorem	eusn 4265*	Two ways to express "𝐴 is a singleton." (Contributed by NM, 30-Oct-2010.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	rabsnt 4266*	Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓)
Theorem	prcom 4267	Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
Theorem	preq1 4268	Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
Theorem	preq2 4269	Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
Theorem	preq12 4270	Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
Theorem	preq1i 4271	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐴, 𝐶} = {𝐵, 𝐶}
Theorem	preq2i 4272	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐶, 𝐴} = {𝐶, 𝐵}
Theorem	preq12i 4273	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ {𝐴, 𝐶} = {𝐵, 𝐷}
Theorem	preq1d 4274	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
Theorem	preq2d 4275	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})
Theorem	preq12d 4276	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})
Theorem	tpeq1 4277	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
Theorem	tpeq2 4278	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
Theorem	tpeq3 4279	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
Theorem	tpeq1d 4280	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
Theorem	tpeq2d 4281	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
Theorem	tpeq3d 4282	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
Theorem	tpeq123d 4283	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐸 = 𝐹)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
Theorem	tprot 4284	Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
Theorem	tpcoma 4285	Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
Theorem	tpcomb 4286	Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
Theorem	tpass 4287	Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
Theorem	qdass 4288	Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})
Theorem	qdassr 4289	Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷})
Theorem	tpidm12 4290	Unordered triple {𝐴, 𝐴, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}
Theorem	tpidm13 4291	Unordered triple {𝐴, 𝐵, 𝐴} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
Theorem	tpidm23 4292	Unordered triple {𝐴, 𝐵, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
Theorem	tpidm 4293	Unordered triple {𝐴, 𝐴, 𝐴} is just an overlong way to write {𝐴}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐴, 𝐴} = {𝐴}
Theorem	tppreq3 4294	An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Theorem	prid1g 4295	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
Theorem	prid2g 4296	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})
Theorem	prid1 4297	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 24-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ {𝐴, 𝐵}
Theorem	prid2 4298	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ 𝐵 ∈ {𝐴, 𝐵}
Theorem	ifpprsnss 4299	An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021.)
⊢ (𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))
Theorem	prprc1 4300	A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})