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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdedth3v 4001 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4000. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ta   =>    |-  ( ph  ->  ps )
 
Theoremdedth4v 4002 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4000. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  S )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  T )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  D ,  U )  ->  ( ta  <->  et ) )   &    |-  et   =>    |-  ( ph  ->  ps )
 
Theoremelimhyp 4003 Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3996. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <->  ps ) )   &    |-  ch   =>    |-  ps
 
Theoremelimhyp2v 4004 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et ) )   &    |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th ) )   &    |-  ta   =>    |-  th
 
Theoremelimhyp3v 4005 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta ) )   &    |-  et   =>    |-  ta
 
Theoremelimhyp4v 4006 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3996). (Contributed by NM, 16-Apr-2005.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( F  =  if ( ph ,  F ,  G )  ->  ( ta  <->  ps ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  rh ) )   &    |-  ( G  =  if ( ph ,  F ,  G )  ->  ( rh  <->  ps ) )   &    |-  et   =>    |-  ps
 
Theoremelimel 4007 Eliminate a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 15-May-1999.)
 |-  B  e.  C   =>    |-  if ( A  e.  C ,  A ,  B )  e.  C
 
Theoremelimdhyp 4008 Version of elimhyp 4003 where the hypothesis is deduced from the final antecedent. See ghomgrplem 29204 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
 |-  ( ph  ->  ps )   &    |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( th  <->  ch ) )   &    |-  th   =>    |-  ch
 
Theoremkeephyp 4009 Transform a hypothesis  ps that we want to keep (but contains the same class variable  A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  ps   &    |-  ch   =>    |-  th
 
Theoremkeephyp2v 4010 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3996). (Contributed by NM, 16-Apr-2005.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et ) )   &    |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th ) )   &    |-  ps   &    |-  ta   =>    |-  th
 
Theoremkeephyp3v 4011 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( rh  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta ) )   &    |-  rh   &    |-  et   =>    |-  ta
 
Theoremkeepel 4012 Keep a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 14-Aug-1999.)
 |-  A  e.  C   &    |-  B  e.  C   =>    |- 
 if ( ph ,  A ,  B )  e.  C
 
Theoremifex 4013 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 if ( ph ,  A ,  B )  e.  _V
 
Theoremifexg 4014 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
 
2.1.16  Power classes
 
Syntaxcpw 4015 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
 class  ~P A
 
Theorempwjust 4016* Soundness justification theorem for df-pw 4017. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  C_  A }  =  {
 y  |  y  C_  A }
 
Definitiondf-pw 4017* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of  _V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if  A  =  { 3 ,  5 ,  7 }, then  ~P A  =  { (/) ,  { 3 } ,  { 5 } ,  { 7 } ,  { 3 ,  5 } ,  { 3 ,  7 } ,  {
5 ,  7 } ,  { 3 ,  5 ,  7 } } (ex-pw 25276). We will later introduce the Axiom of Power Sets ax-pow 4634, which can be expressed in class notation per pwexg 4640. Still later we will prove, in hashpw 12497, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993.)
 |- 
 ~P A  =  { x  |  x  C_  A }
 
Theorempweq 4018 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)
 |-  ( A  =  B  ->  ~P A  =  ~P B )
 
Theorempweqi 4019 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
 |-  A  =  B   =>    |-  ~P A  =  ~P B
 
Theorempweqd 4020 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ~P A  =  ~P B )
 
Theoremelpw 4021 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  ~P B  <->  A  C_  B )
 
Theoremselpw 4022* Setvar variable membership in a power class (common case). See elpw 4021. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( x  e.  ~P A 
 <->  x  C_  A )
 
Theoremelpwg 4023 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4619. (Contributed by NM, 6-Aug-2000.)
 |-  ( A  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpwi 4024 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  ~P B  ->  A  C_  B )
 
Theoremelpwid 4025 An element of a power class is a subclass. Deduction form of elpwi 4024. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ~P B )   =>    |-  ( ph  ->  A 
 C_  B )
 
Theoremelelpwi 4026 If  A belongs to a part of  C then  A belongs to  C. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  e.  B  /\  B  e.  ~P C )  ->  A  e.  C )
 
Theoremnfpw 4027 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x ~P A
 
Theorempwidg 4028 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  ~P A )
 
Theorempwid 4029 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  A  e.  ~P A
 
Theorempwss 4030* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
 |-  ( ~P A  C_  B 
 <-> 
 A. x ( x 
 C_  A  ->  x  e.  B ) )
 
2.1.17  Unordered and ordered pairs
 
Theoremsnjust 4031* Soundness justification theorem for df-sn 4033. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  =  A }  =  {
 y  |  y  =  A }
 
Syntaxcsn 4032 Extend class notation to include singleton.
 class  { A }
 
Definitiondf-sn 4033* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of  _V, although it is not very meaningful in this case. For an alternate definition see dfsn2 4045. (Contributed by NM, 21-Jun-1993.)
 |- 
 { A }  =  { x  |  x  =  A }
 
Syntaxcpr 4034 Extend class notation to include unordered pair.
 class  { A ,  B }
 
Definitiondf-pr 4035 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example,  A  e.  {
1 ,  -u 1 }  ->  ( A ^
2 )  =  1 (ex-pr 25277). They are unordered, so  { A ,  B }  =  { B ,  A } as proven by prcom 4110. For a more traditional definition, but requiring a dummy variable, see dfpr2 4047. (Contributed by NM, 21-Jun-1993.)
 |- 
 { A ,  B }  =  ( { A }  u.  { B } )
 
Syntaxctp 4036 Extend class notation to include unordered triplet.
 class  { A ,  B ,  C }
 
Definitiondf-tp 4037 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
 |- 
 { A ,  B ,  C }  =  ( { A ,  B }  u.  { C }
 )
 
Syntaxcop 4038 Extend class notation to include ordered pair.
 class  <. A ,  B >.
 
Definitiondf-op 4039* 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 4242, opprc2 4243, and 0nelop 4746). For Kuratowski's actual definition when the arguments are sets, see dfop 4218. For the justifying theorem (for sets) see opth 4730. See dfopif 4216 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 4039 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4039 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 4758. 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 8052, 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 5056. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 12352. An ordered pair of real numbers can also be represented by a complex number as shown by cru 10548. 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 4216. (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 4040 Extend class notation to include ordered triple.
 class  <. A ,  B ,  C >.
 
Definitiondf-ot 4041 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 4042 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
 |-  ( A  =  B  ->  { A }  =  { B } )
 
Theoremsneqi 4043 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  A  =  B   =>    |-  { A }  =  { B }
 
Theoremsneqd 4044 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A }  =  { B } )
 
Theoremdfsn2 4045 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
 |- 
 { A }  =  { A ,  A }
 
Theoremelsn 4046* 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 4047* 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 4048 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 )
 ) )
 
Theoremeldifpr 4049 Membership in a set with two elements removed. Similar to eldifsn 4157 and eldiftp 4075. (Contributed by Mario Carneiro, 18-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D } )  <->  ( A  e.  B  /\  A  =/=  C  /\  A  =/=  D ) )
 
Theoremelpr 4050 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 4051 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 )
 )
 
Theoremelpri 4052 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 ) )
 
Theoremnelpri 4053 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 4054 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 } )
 
Theoremelsncg 4055 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 4056 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 4057 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  { B }  ->  A  =  B )
 
Theoremsnidg 4058 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 4059 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 4060 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 4061 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  x  e.  { x }
 
Theoremelsnc2g 4062 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 4063 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 4064* 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 4065* 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 4066* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 4065 instead. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsng 4067* 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 4068* 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 4069* 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 4070 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 4071* 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 4072* 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 4073 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 4074 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 4075 Membership in a set with three elements removed. Similar to eldifsn 4157 and eldifpr 4049. (Contributed by David A. Wheeler, 22-Jul-2017.)
 |-  ( A  e.  ( B  \  { C ,  D ,  E }
 ) 
 <->  ( A  e.  B  /\  ( A  =/=  C  /\  A  =/=  D  /\  A  =/=  E ) ) )
 
Theoremeltpi 4076 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 4077 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 4078* 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 4079 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x { A ,  B }
 
Theoremifpr 4080 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 4081* 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 4082* 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 4083* 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 4084* 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 4085* 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 4086* 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 4087* 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 4088* 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 4089 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   =>    |-  F/_ x { A }
 
Theoremcsbsng 4090 Distribute proper substitution through the singleton of a class. csbsng 4090 is derived from the virtual deduction proof csbsngVD 33794. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { B }  =  { [_ A  /  x ]_ B } )
 
Theoremcsbprg 4091 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 4092 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 4093 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
 |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
 
Theoremdisjpr2 4094 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 4095 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 4096 A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
 |-  ( A  e.  _V  <->  { A }  =/=  (/) )
 
Theoremr19.12sn 4097* Special case of r19.12 2983 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 4098* Older form of r19.12sn 4097. Obsolete as of 18-Mar-2020. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph )
 
Theoremrabsn 4099* 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 4100* 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 } ,  (/) )
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