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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssnid 4001 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  x  e.  { x }
 
Theoremelsnc2g 4002 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 4003 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 4004* 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 4005* 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 4006* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Deletion blocked by r19.12snOLD 4038. Use rexsns 4005 instead. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsng 4007* 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 4008* 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 4009* 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 4010 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 4011* 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 4012* 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 4013 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 )
 
Theoremeltpg 4014 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 4015 Membership in a set with three elements removed. Similar to eldifsn 4097 and eldifpr 3989. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E }
 ) 
 <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremeltpi 4016 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 4017 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 4018* 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 4019 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x { A ,  B }
 
Theoremifpr 4020 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 4021* 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 4022* 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 4023* 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 4024* 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 4025* 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 4026* 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 4027* 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 4028* 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 4029 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   =>    |-  F/_ x { A }
 
Theoremcsbsng 4030 Distribute proper substitution through the singleton of a class. csbsng 4030 is derived from the virtual deduction proof csbsngVD 36724. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { B }  =  { [_ A  /  x ]_ B } )
 
Theoremcsbprg 4031 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 4032 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 4033 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
 |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
 
Theoremdisjpr2 4034 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 4035 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 4036 A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
 |-  ( A  e.  _V  <->  { A }  =/=  (/) )
 
Theoremr19.12sn 4037* Special case of r19.12 2933 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 4038* Older form of r19.12sn 4037. 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 4039* 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 4040* 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 4041* 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 4042* 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 4043* 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 4044 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 4045* 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 4046 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
 |-  ( ( A  e.  V  /\  { x  |  ph
 }  =  { A } )  ->  E! x ph )
 
Theoremrabsneu 4047 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 4048* Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)
 |-  ( E! x  x  e.  A  <->  E. x  A  =  { x } )
 
Theoremrabsnt 4049* 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 4050 Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
 |- 
 { A ,  B }  =  { B ,  A }
 
Theorempreq1 4051 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
 |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C } )
 
Theorempreq2 4052 Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)
 |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B } )
 
Theorempreq12 4053 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D }
 )
 
Theorempreq1i 4054 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { A ,  C }  =  { B ,  C }
 
Theorempreq2i 4055 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { C ,  A }  =  { C ,  B }
 
Theorempreq12i 4056 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 { A ,  C }  =  { B ,  D }
 
Theorempreq1d 4057 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  C }
 )
 
Theorempreq2d 4058 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A }  =  { C ,  B }
 )
 
Theorempreq12d 4059 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
 
Theoremtpeq1 4060 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2 4061 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3 4062 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq1d 4063 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2d 4064 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3d 4065 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq123d 4066 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 4067 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 { A ,  B ,  C }  =  { B ,  C ,  A }
 
Theoremtpcoma 4068 Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { B ,  A ,  C }
 
Theoremtpcomb 4069 Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { A ,  C ,  B }
 
Theoremtpass 4070 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 { A ,  B ,  C }  =  ( { A }  u.  { B ,  C }
 )
 
Theoremqdass 4071 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 4072 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 4073 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 4074 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 4075 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 4076 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 4077 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 4078 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
 |-  ( A  e.  V  ->  A  e.  { A ,  B } )
 
Theoremprid2g 4079 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
 |-  ( B  e.  V  ->  B  e.  { A ,  B } )
 
Theoremprid1 4080 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 24-Jun-1993.)
 |-  A  e.  _V   =>    |-  A  e.  { A ,  B }
 
Theoremprid2 4081 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  B  e.  _V   =>    |-  B  e.  { A ,  B }
 
Theoremprprc1 4082 A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
 |-  ( -.  A  e.  _V 
 ->  { A ,  B }  =  { B } )
 
Theoremprprc2 4083 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
 |-  ( -.  B  e.  _V 
 ->  { A ,  B }  =  { A } )
 
Theoremprprc 4084 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
 |-  ( ( -.  A  e.  _V  /\  -.  B  e.  _V )  ->  { A ,  B }  =  (/) )
 
Theoremtpid1 4085 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   =>    |-  A  e.  { A ,  B ,  C }
 
Theoremtpid2 4086 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  B  e.  _V   =>    |-  B  e.  { A ,  B ,  C }
 
Theoremtpid3g 4087 Closed theorem form of tpid3 4088. This proof was automatically generated from the virtual deduction proof tpid3gVD 36672 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremtpid3 4088 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  C  e.  _V   =>    |-  C  e.  { A ,  B ,  C }
 
Theoremsnnzg 4089 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  e.  V  ->  { A }  =/=  (/) )
 
Theoremsnnz 4090 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A }  =/= 
 (/)
 
Theoremprnz 4091 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B }  =/=  (/)
 
Theoremprnzg 4092 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
 
Theoremtpnz 4093 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B ,  C }  =/= 
 (/)
 
Theoremtpnzd 4094 A triplet containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)
 |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  { A ,  B ,  C }  =/= 
 (/) )
 
Theoremraltpd 4095* Convert a quantification over a triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
 |-  ( ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   &    |-  (
 ( ph  /\  x  =  B )  ->  ( ps 
 <-> 
 th ) )   &    |-  (
 ( ph  /\  x  =  C )  ->  ( ps 
 <->  ta ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  (
 A. x  e.  { A ,  B ,  C } ps  <->  ( ch  /\  th 
 /\  ta ) ) )
 
Theoremsnss 4096 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <->  { A }  C_  B )
 
Theoremeldifsn 4097 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
 |-  ( A  e.  ( B  \  { C }
 ) 
 <->  ( A  e.  B  /\  A  =/=  C ) )
 
Theoremeldifsni 4098 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
 |-  ( A  e.  ( B  \  { C }
 )  ->  A  =/=  C )
 
Theoremneldifsn 4099  A is not in  ( B 
\  { A }
). (Contributed by David Moews, 1-May-2017.)
 |- 
 -.  A  e.  ( B  \  { A }
 )
 
Theoremneldifsnd 4100  A is not in  ( B 
\  { A }
). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  A  e.  ( B  \  { A } ) )
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