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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelsn2g 4201 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.)
(𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremelsn2 4202 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)
 
Theoremnelsn 4203 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.)
(𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
 
TheoremnelsnOLD 4204 Obsolete proof of nelsn 4203 as of 4-May-2021. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
 
Theoremrabeqsn 4205* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))
 
Theoremrabsssn 4206* 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.)
({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
 
Theoremralsnsg 4207* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 
Theoremrexsns 4208* 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)
 
Theoremralsng 4209* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
 
Theoremrexsng 4210* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
 
Theorem2ralsng 4211* Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))
 
Theoremexsnrex 4212 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} ↔ ∃𝑥𝑀 𝑀 = {𝑥})
 
Theoremralsn 4213* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥 ∈ {𝐴}𝜑𝜓)
 
Theoremrexsn 4214* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥 ∈ {𝐴}𝜑𝜓)
 
Theoremelpwunsn 4215 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
(𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶𝐴)
 
Theoremeqoreldif 4216 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.)
(𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
 
TheoremeqoreldifOLD 4217 Obsolete proof of eqoreldif 4216 as of 23-Jul-2021. (Contributed by AV, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
 
Theoremeltpg 4218 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
 
Theoremeldiftp 4219 Membership in a set with three elements removed. Similar to eldifsn 4308 and eldifpr 4195. (Contributed by David A. Wheeler, 22-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
 
Theoremeltpi 4220 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))
 
Theoremeltp 4221 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 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))
 
Theoremdftp2 4222* 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 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
 
Theoremnfpr 4223 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥{𝐴, 𝐵}
 
Theoremifpr 4224 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
((𝐴𝐶𝐵𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
 
Theoremralprg 4225* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 
Theoremrexprg 4226* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 
Theoremraltpg 4227* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
 
Theoremrextpg 4228* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
 
Theoremralpr 4229* 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
 
Theoremrexpr 4230* 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
 
Theoremraltp 4231* 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
 
Theoremrextp 4232* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
 
Theoremnfsn 4233 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴       𝑥{𝐴}
 
Theoremcsbsng 4234 Distribute proper substitution through the singleton of a class. csbsng 4234 is derived from the virtual deduction proof csbsngVD 38949. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 
Theoremcsbprg 4235 Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
(𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})
 
Theoremelinsn 4236 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.)
((𝐴𝑉 ∧ (𝐵𝐶) = {𝐴}) → (𝐴𝐵𝐴𝐶))
 
Theoremdisjsn 4237 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}) = ∅ ↔ ¬ 𝐵𝐴)
 
Theoremdisjsn2 4238 Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998.)
(𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
 
Theoremdisjpr2 4239 Two completely distinct unordered pairs are disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Proof shortened by JJ, 23-Jul-2021.)
(((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
 
Theoremdisjpr2OLD 4240 Obsolete proof of disjpr2 4239 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
 
Theoremdisjprsn 4241 The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
 
Theoremdisjtpsn 4242 The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.)
((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)
 
Theoremdisjtp2 4243 Two completely distinct unordered triples are disjoint. (Contributed by AV, 14-Nov-2021.)
(((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)
 
Theoremsnprc 4244 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} = ∅)
 
Theoremsnnzb 4245 A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
(𝐴 ∈ V ↔ {𝐴} ≠ ∅)
 
Theoremr19.12sn 4246* Special case of r19.12 3059 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
(𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
 
Theoremrabsn 4247* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
(𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
 
Theoremrabsnifsb 4248* 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}, ∅)
 
Theoremrabsnif 4249* 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 → (𝜑𝜓))       {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
 
Theoremrabrsn 4250* 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}))
 
Theoremeuabsn2 4251* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
 
Theoremeuabsn 4252 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
(∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
 
Theoremreusn 4253* 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.)
(∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
 
Theoremabsneu 4254 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
 
Theoremrabsneu 4255 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
 
Theoremeusn 4256* Two ways to express "𝐴 is a singleton." (Contributed by NM, 30-Oct-2010.)
(∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremrabsnt 4257* 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 → (𝜑𝜓))       ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
 
Theoremprcom 4258 Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
{𝐴, 𝐵} = {𝐵, 𝐴}
 
Theorempreq1 4259 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
(𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
 
Theorempreq2 4260 Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)
(𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
 
Theorempreq12 4261 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
 
Theorempreq1i 4262 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐴, 𝐶} = {𝐵, 𝐶}
 
Theorempreq2i 4263 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐶, 𝐴} = {𝐶, 𝐵}
 
Theorempreq12i 4264 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       {𝐴, 𝐶} = {𝐵, 𝐷}
 
Theorempreq1d 4265 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
 
Theorempreq2d 4266 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})
 
Theorempreq12d 4267 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})
 
Theoremtpeq1 4268 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
 
Theoremtpeq2 4269 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 
Theoremtpeq3 4270 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 
Theoremtpeq1d 4271 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
 
Theoremtpeq2d 4272 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 
Theoremtpeq3d 4273 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 
Theoremtpeq123d 4274 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐸 = 𝐹)       (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
 
Theoremtprot 4275 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
 
Theoremtpcoma 4276 Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
 
Theoremtpcomb 4277 Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
 
Theoremtpass 4278 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
{𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
 
Theoremqdass 4279 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})
 
Theoremqdassr 4280 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷})
 
Theoremtpidm12 4281 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}
 
Theoremtpidm13 4282 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}
 
Theoremtpidm23 4283 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}
 
Theoremtpidm 4284 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}
 
Theoremtppreq3 4285 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})
 
Theoremprid1g 4286 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
(𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
 
Theoremprid2g 4287 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
(𝐵𝑉𝐵 ∈ {𝐴, 𝐵})
 
Theoremprid1 4288 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}
 
Theoremprid2 4289 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}
 
Theoremifpprsnss 4290 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))
 
Theoremprprc1 4291 A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
 
Theoremprprc2 4292 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
 
Theoremprprc 4293 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
 
Theoremtpid1 4294 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V       𝐴 ∈ {𝐴, 𝐵, 𝐶}
 
Theoremtpid2 4295 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐵 ∈ V       𝐵 ∈ {𝐴, 𝐵, 𝐶}
 
Theoremtpid3g 4296 Closed theorem form of tpid3 4298. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremtpid3gOLD 4297 Obsolete proof of tpid3g 4296 as of 30-Apr-2021. Closed theorem form of tpid3 4298. This proof was automatically generated from the virtual deduction proof tpid3gVD 38897 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremtpid3 4298 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 30-Apr-2021.)
𝐶 ∈ V       𝐶 ∈ {𝐴, 𝐵, 𝐶}
 
Theoremsnnzg 4299 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
(𝐴𝑉 → {𝐴} ≠ ∅)
 
Theoremsnnz 4300 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴} ≠ ∅
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