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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelsni 3601 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
 
Theoremdfpr2 3602* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
 
Theoremelprg 3603 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 
Theoremelpr 3604 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpr2 3605 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpri 3606 If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
(𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremnelpri 3607 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.)
𝐴𝐵    &   𝐴𝐶        ¬ 𝐴 ∈ {𝐵, 𝐶}
 
Theoremprneli 3608 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.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∉ {𝐵, 𝐶}
 
Theoremnelprd 3609 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.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
 
Theoremeldifpr 3610 Membership in a set with two elements removed. Similar to eldifsn 3710 and eldiftp 3629. (Contributed by Mario Carneiro, 18-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
 
Theoremrexdifpr 3611 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
 
Theoremsnidg 3612 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
(𝐴𝑉𝐴 ∈ {𝐴})
 
Theoremsnidb 3613 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
 
Theoremsnid 3614 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}
 
Theoremvsnid 3615 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
𝑥 ∈ {𝑥}
 
Theoremelsn2g 3616 There is only 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 3617 There is only 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 3618 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})
 
Theoremmosn 3619* A singleton has at most one element. This works whether 𝐴 is a proper class or not, and in that sense can be seen as encompassing both snmg 3701 and snprc 3648. (Contributed by Jim Kingdon, 30-Aug-2018.)
∃*𝑥 𝑥 ∈ {𝐴}
 
Theoremralsnsg 3620* 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))
 
Theoremralsns 3621* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 
Theoremrexsns 3622* 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 3623* 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 3624* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
 
Theoremexsnrex 3625 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 3626* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥 ∈ {𝐴}𝜑𝜓)
 
Theoremrexsn 3627* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥 ∈ {𝐴}𝜑𝜓)
 
Theoremeltpg 3628 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 3629 Membership in a set with three elements removed. Similar to eldifsn 3710 and eldifpr 3610. (Contributed by David A. Wheeler, 22-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
 
Theoremeltpi 3630 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 3631 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 3632* 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 3633 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥{𝐴, 𝐵}
 
Theoremralprg 3634* 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 3635* 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 3636* 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 3637* 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 3638* 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 3639* 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 3640* 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 3641* 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))
 
Theoremsbcsng 3642* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝐴}𝜑))
 
Theoremnfsn 3643 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴       𝑥{𝐴}
 
Theoremcsbsng 3644 Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 
Theoremdisjsn 3645 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 3646 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
(𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
 
Theoremdisjpr2 3647 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
(((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
 
Theoremsnprc 3648 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, 5-Aug-1993.)
𝐴 ∈ V ↔ {𝐴} = ∅)
 
Theoremr19.12sn 3649* Special case of r19.12 2576 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
(𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
 
Theoremrabsn 3650* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
(𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
 
Theoremrabrsndc 3651* A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
𝐴 ∈ V    &   DECID 𝜑       (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
 
Theoremeuabsn2 3652* 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 3653 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 3654* 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 3655 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
 
Theoremrabsneu 3656 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 3657* Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.)
(∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremrabsnt 3658* 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 3659 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
{𝐴, 𝐵} = {𝐵, 𝐴}
 
Theorempreq1 3660 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
(𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
 
Theorempreq2 3661 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
 
Theorempreq12 3662 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
 
Theorempreq1i 3663 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐴, 𝐶} = {𝐵, 𝐶}
 
Theorempreq2i 3664 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐶, 𝐴} = {𝐶, 𝐵}
 
Theorempreq12i 3665 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       {𝐴, 𝐶} = {𝐵, 𝐷}
 
Theorempreq1d 3666 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
 
Theorempreq2d 3667 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})
 
Theorempreq12d 3668 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})
 
Theoremtpeq1 3669 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
 
Theoremtpeq2 3670 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 
Theoremtpeq3 3671 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 
Theoremtpeq1d 3672 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
 
Theoremtpeq2d 3673 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 
Theoremtpeq3d 3674 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 
Theoremtpeq123d 3675 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 3676 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
 
Theoremtpcoma 3677 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
 
Theoremtpcomb 3678 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
 
Theoremtpass 3679 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
{𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
 
Theoremqdass 3680 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})
 
Theoremqdassr 3681 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷})
 
Theoremtpidm12 3682 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 3683 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 3684 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 3685 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 3686 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 3687 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 3688 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 3689 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝐴 ∈ {𝐴, 𝐵}
 
Theoremprid2 3690 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}
 
Theoremprprc1 3691 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
 
Theoremprprc2 3692 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
 
Theoremprprc 3693 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
 
Theoremtpid1 3694 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}
 
Theoremtpid1g 3695 Closed theorem form of tpid1 3694. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})
 
Theoremtpid2 3696 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}
 
Theoremtpid2g 3697 Closed theorem form of tpid2 3696. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
 
Theoremtpid3g 3698 Closed theorem form of tpid3 3699. (Contributed by Alan Sare, 24-Oct-2011.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremtpid3 3699 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐶 ∈ V       𝐶 ∈ {𝐴, 𝐵, 𝐶}
 
Theoremsnnzg 3700 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
(𝐴𝑉 → {𝐴} ≠ ∅)
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