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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-tp 4001 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
 |- 
 { A ,  B ,  C }  =  ( { A ,  B }  u.  { C }
 )
 
Syntaxcop 4002 Extend class notation to include ordered pair.
 class  <. A ,  B >.
 
Definitiondf-op 4003* Definition of an ordered pair, equivalent to Kuratowski's definition  { { A } ,  { A ,  B } } when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 4207, opprc2 4208, and 0nelop 4707). For Kuratowski's actual definition when the arguments are sets, see dfop 4183. For the justifying theorem (for sets) see opth 4692. See dfopif 4181 for an equivalent formulation using the  if operation.

Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as  <. A ,  B >.  =  { { A } ,  { A ,  B } }, which has different behavior from our df-op 4003 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4003 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses.

There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition  <. A ,  B >._2  =  { { { A } ,  (/) } ,  { { B } } }, justified by opthwiener 4719. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition  <. A ,  B >._3  =  { A ,  { A ,  B } } is justified by opthreg 8126, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is  <. A ,  B >._4  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { { (/) } } ) ), justified by opthprc 4898. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 12456. An ordered pair of real numbers can also be represented by a complex number as shown by cru 10602. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternative definition in New Foundations is the Definition from [Rosser] p. 281.

Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4181. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

 |- 
 <. A ,  B >.  =  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
 
Syntaxcotp 4004 Extend class notation to include ordered triple.
 class  <. A ,  B ,  C >.
 
Definitiondf-ot 4005 Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
 |- 
 <. A ,  B ,  C >.  =  <. <. A ,  B >. ,  C >.
 
Theoremsneq 4006 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
 |-  ( A  =  B  ->  { A }  =  { B } )
 
Theoremsneqi 4007 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  A  =  B   =>    |-  { A }  =  { B }
 
Theoremsneqd 4008 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A }  =  { B } )
 
Theoremdfsn2 4009 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
 |- 
 { A }  =  { A ,  A }
 
Theoremelsn 4010* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
 |-  ( x  e.  { A }  <->  x  =  A )
 
Theoremdfpr2 4011* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
 |- 
 { A ,  B }  =  { x  |  ( x  =  A  \/  x  =  B ) }
 
Theoremelprg 4012 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.)
 |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
 ) )
 
Theoremelpri 4013 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  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
 
Theoremelpr 4014 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  e.  _V   =>    |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
 )
 
Theoremelpr2 4015 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  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
 )
 
Theoremnelpri 4016 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  e.  { B ,  C }
 
Theoremnelprd 4017 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.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  -.  A  e.  { B ,  C } )
 
Theoremeldifpr 4018 Membership in a set with two elements removed. Similar to eldifsn 4122 and eldiftp 4040. (Contributed by Mario Carneiro, 18-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D } )  <->  ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D ) )
 
Theoremelsncg 4019 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( A  e.  { B }  <->  A  =  B ) )
 
Theoremelsnc 4020 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
 |-  A  e.  _V   =>    |-  ( A  e.  { B }  <->  A  =  B )
 
Theoremelsni 4021 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  { B }  ->  A  =  B )
 
Theoremsnidg 4022 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
 |-  ( A  e.  V  ->  A  e.  { A } )
 
Theoremsnidb 4023 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
 |-  ( A  e.  _V  <->  A  e.  { A } )
 
Theoremsnid 4024 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  A  e.  { A }
 
Theoremssnid 4025 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  x  e.  { x }
 
Theoremelsnc2g 4026 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.)
 |-  ( B  e.  V  ->  ( A  e.  { B }  <->  A  =  B ) )
 
Theoremelsnc2 4027 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  e.  _V   =>    |-  ( A  e.  { B }  <->  A  =  B )
 
Theoremralsnsg 4028* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( A. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremrexsns 4029* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
 |-  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 
TheoremrexsnsOLD 4030* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Deletion blocked by r19.12snOLD 4063. Use rexsns 4029 instead. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsng 4031* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
 
Theoremrexsng 4032* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
 
Theorem2ralsng 4033* Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e.  { A } A. y  e.  { B } ph  <->  ch ) )
 
Theoremexsnrex 4034 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.)
 |-  ( E. x  M  =  { x }  <->  E. x  e.  M  M  =  { x } )
 
Theoremralsn 4035* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  { A } ph  <->  ps )
 
Theoremrexsn 4036* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  { A } ph  <->  ps )
 
Theoremelpwunsn 4037 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
 |-  ( A  e.  ( ~P ( B  u.  { C } )  \  ~P B )  ->  C  e.  A )
 
Theoremeqoreldif 4038 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.)
 |-  ( B  e.  C  ->  ( A  e.  C  <->  ( A  =  B  \/  A  e.  ( C  \  { B } )
 ) ) )
 
Theoremeltpg 4039 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
 |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) ) )
 
Theoremeldiftp 4040 Membership in a set with three elements removed. Similar to eldifsn 4122 and eldifpr 4018. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E }
 ) 
 <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremeltpi 4041 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( A  e.  { B ,  C ,  D }  ->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
 
Theoremeltp 4042 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  e.  _V   =>    |-  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
 
Theoremdftp2 4043* 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 ) }
 
Theoremnfpr 4044 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x { A ,  B }
 
Theoremifpr 4045 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  if ( ph ,  A ,  B )  e.  { A ,  B } )
 
Theoremralprg 4046* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e.  { A ,  B } ph  <->  ( ps  /\  ch ) ) )
 
Theoremrexprg 4047* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e.  { A ,  B } ph  <->  ( ps  \/  ch ) ) )
 
Theoremraltpg 4048* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e.  { A ,  B ,  C } ph 
 <->  ( ps  /\  ch  /\ 
 th ) ) )
 
Theoremrextpg 4049* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  e.  { A ,  B ,  C } ph 
 <->  ( ps  \/  ch  \/  th ) ) )
 
Theoremralpr 4050* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( A. x  e. 
 { A ,  B } ph  <->  ( ps  /\  ch ) )
 
Theoremrexpr 4051* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( E. x  e. 
 { A ,  B } ph  <->  ( ps  \/  ch ) )
 
Theoremraltp 4052* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( A. x  e.  { A ,  B ,  C } ph  <->  ( ps  /\  ch 
 /\  th ) )
 
Theoremrextp 4053* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  C  ->  (
 ph 
 <-> 
 th ) )   =>    |-  ( E. x  e.  { A ,  B ,  C } ph  <->  ( ps  \/  ch 
 \/  th ) )
 
Theoremnfsn 4054 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   =>    |-  F/_ x { A }
 
Theoremcsbsng 4055 Distribute proper substitution through the singleton of a class. csbsng 4055 is derived from the virtual deduction proof csbsngVD 37151. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { B }  =  { [_ A  /  x ]_ B } )
 
Theoremcsbprg 4056 Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
 |-  ( C  e.  V  -> 
 [_ C  /  x ]_
 { A ,  B }  =  { [_ C  /  x ]_ A ,  [_ C  /  x ]_ B } )
 
Theoremdisjsn 4057 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  i^i  { B } )  =  (/) 
 <->  -.  B  e.  A )
 
Theoremdisjsn2 4058 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
 |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
 
Theoremdisjpr2 4059 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  ( ( ( A  =/=  C  /\  B  =/=  C )  /\  ( A  =/=  D  /\  B  =/=  D ) )  ->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
 
Theoremsnprc 4060 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  e.  _V  <->  { A }  =  (/) )
 
Theoremsnnzb 4061 A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
 |-  ( A  e.  _V  <->  { A }  =/=  (/) )
 
Theoremr19.12sn 4062* Special case of r19.12 2954 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph ) )
 
Theoremr19.12snOLD 4063* Older form of r19.12sn 4062. Obsolete as of 18-Mar-2020. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph )
 
Theoremrabsn 4064* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
 |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
 
Theoremrabsnifsb 4065* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
 |- 
 { x  e.  { A }  |  ph }  =  if ( [. A  /  x ]. ph ,  { A } ,  (/) )
 
Theoremrabsnif 4066* 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  ->  ( ph  <->  ps ) )   =>    |-  { x  e. 
 { A }  |  ph
 }  =  if ( ps ,  { A } ,  (/) )
 
Theoremrabrsn 4067* 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  e.  { A }  |  ph }  ->  ( M  =  (/)  \/  M  =  { A } )
 )
 
Theoremeuabsn2 4068* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y }
 )
 
Theoremeuabsn 4069 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
 |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x }
 )
 
Theoremreusn 4070* 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.)
 |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
 
Theoremabsneu 4071 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
 |-  ( ( A  e.  V  /\  { x  |  ph
 }  =  { A } )  ->  E! x ph )
 
Theoremrabsneu 4072 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  ( ( A  e.  V  /\  { x  e.  B  |  ph }  =  { A } )  ->  E! x  e.  B  ph )
 
Theoremeusn 4073* Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)
 |-  ( E! x  x  e.  A  <->  E. x  A  =  { x } )
 
Theoremrabsnt 4074* 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  e.  _V   &    |-  ( x  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
 
Theoremprcom 4075 Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
 |- 
 { A ,  B }  =  { B ,  A }
 
Theorempreq1 4076 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
 |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C } )
 
Theorempreq2 4077 Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)
 |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B } )
 
Theorempreq12 4078 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D }
 )
 
Theorempreq1i 4079 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { A ,  C }  =  { B ,  C }
 
Theorempreq2i 4080 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { C ,  A }  =  { C ,  B }
 
Theorempreq12i 4081 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 { A ,  C }  =  { B ,  D }
 
Theorempreq1d 4082 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  C }
 )
 
Theorempreq2d 4083 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A }  =  { C ,  B }
 )
 
Theorempreq12d 4084 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
 
Theoremtpeq1 4085 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2 4086 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3 4087 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq1d 4088 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2d 4089 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3d 4090 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq123d 4091 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   &    |-  ( ph  ->  E  =  F )   =>    |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )
 
Theoremtprot 4092 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 { A ,  B ,  C }  =  { B ,  C ,  A }
 
Theoremtpcoma 4093 Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { B ,  A ,  C }
 
Theoremtpcomb 4094 Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { A ,  C ,  B }
 
Theoremtpass 4095 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 { A ,  B ,  C }  =  ( { A }  u.  { B ,  C }
 )
 
Theoremqdass 4096 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B ,  C }  u.  { D } )
 
Theoremqdassr 4097 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A }  u.  { B ,  C ,  D } )
 
Theoremtpidm12 4098 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 4099 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 4100 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 }
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