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Theorem List for Metamath Proof Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-pw 3601* 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 20760). We will later introduce the Axiom of Power Sets ax-pow 4160, which can be expressed in class notation per pwexg 4166. Still later we will prove, in hashpw 11354, 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, 5-Aug-1993.)
 |- 
 ~P A  =  { x  |  x  C_  A }
 
Theorempweq 3602 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ~P A  =  ~P B )
 
Theorempweqi 3603 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
 |-  A  =  B   =>    |-  ~P A  =  ~P B
 
Theorempweqd 3604 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ~P A  =  ~P B )
 
Theoremelpw 3605 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 )
 
Theoremelpwg 3606 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4141. (Contributed by NM, 6-Aug-2000.)
 |-  ( A  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpwi 3607 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  ~P B  ->  A  C_  B )
 
Theoremelpwid 3608 An element of a power class is a subclass. Deduction form of elpwi 3607. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ~P B )   =>    |-  ( ph  ->  A 
 C_  B )
 
Theoremelelpwi 3609 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 3610 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 3611 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  ~P A )
 
Theorempwid 3612 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 3613* 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
 
Syntaxcsn 3614 Extend class notation to include singleton.
 class  { A }
 
Syntaxcpr 3615 Extend class notation to include unordered pair.
 class  { A ,  B }
 
Syntaxctp 3616 Extend class notation to include unordered triplet.
 class  { A ,  B ,  C }
 
Syntaxcop 3617 Extend class notation to include ordered pair.
 class  <. A ,  B >.
 
Syntaxcotp 3618 Extend class notation to include ordered triple.
 class  <. A ,  B ,  C >.
 
Theoremsnjust 3619* Soundness justification theorem for df-sn 3620. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  =  A }  =  {
 y  |  y  =  A }
 
Definitiondf-sn 3620* 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 3628. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A }  =  { x  |  x  =  A }
 
Definitiondf-pr 3621 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example,  A  e.  {
1 ,  -u 1 }  ->  ( A ^
2 )  =  1 (ex-pr 20761). They are unordered, so  { A ,  B }  =  { B ,  A } as proven by prcom 3679. For a more traditional definition, but requiring a dummy variable, see dfpr2 3630. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  =  ( { A }  u.  { B } )
 
Definitiondf-tp 3622 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
 |- 
 { A ,  B ,  C }  =  ( { A ,  B }  u.  { C }
 )
 
Definitiondf-op 3623* 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 3792, opprc2 3793, and 0nelop 4228). For Kuratowski's actual definition when the arguments are sets, see dfop 3769. For the justifying theorem (for sets) see opth 4217. See dfopif 3767 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 3623 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3623 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 4240. 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 7287, 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 4724. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 11252. Finally, an ordered pair of real numbers can be represented by a complex number as shown by cru 9706. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

 |- 
 <. A ,  B >.  =  { x  |  ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
 
Definitiondf-ot 3624 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 3625 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  { A }  =  { B } )
 
Theoremsneqi 3626 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  A  =  B   =>    |-  { A }  =  { B }
 
Theoremsneqd 3627 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A }  =  { B } )
 
Theoremdfsn2 3628 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
 |- 
 { A }  =  { A ,  A }
 
Theoremelsn 3629* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  { A }  <->  x  =  A )
 
Theoremdfpr2 3630* 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 3631 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 )
 ) )
 
Theoremelpr 3632 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 3633 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 3634 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 3635 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 }
 
Theoremelsncg 3636 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 3637 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 3638 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  { B }  ->  A  =  B )
 
Theoremsnidg 3639 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 3640 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 3641 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 }
 
Theoremelsnc2g 3642 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 3643 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 )
 
Theoremralsns 3644* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( A. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremrexsns 3645* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
 |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  [. A  /  x ]. ph )
 )
 
Theoremralsng 3646* 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 3647* 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 ) )
 
Theoremralsn 3648* 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 3649* 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 )
 
Theoremeltpg 3650 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 ) ) )
 
Theoremeltpi 3651 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 3652 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 3653* 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 3654 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x { A ,  B }
 
Theoremifpr 3655 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 3656* 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 3657* 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 3658* 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 3659* 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 3660* 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 3661* 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 3662* 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 3663* 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 ) )
 
Theoremsbcsng 3664* 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 ]. ph  <->  A. x  e.  { A } ph ) )
 
Theoremnfsn 3665 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   =>    |-  F/_ x { A }
 
Theoremcsbsng 3666 Distribute proper substitution through the singleton of a class. csbsng 3666 is derived from the virtual deduction proof csbsngVD 27802. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 { B }  =  { [_ A  /  x ]_ B } )
 
Theoremdisjsn 3667 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 3668 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
 |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
 
Theoremsnprc 3669 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  e.  _V  <->  { A }  =  (/) )
 
Theoremr19.12sn 3670* Special case of r19.12 2631 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  { A } A. y  e.  B  ph  <->  A. y  e.  B  E. x  e.  { A } ph )
 
Theoremrabsn 3671* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
 |-  ( B  e.  A  ->  { x  e.  A  |  x  =  B }  =  { B } )
 
Theoremeuabsn2 3672* 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 3673 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 3674* 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 3675 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
 |-  ( ( A  e.  V  /\  { x  |  ph
 }  =  { A } )  ->  E! x ph )
 
Theoremrabsneu 3676 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 3677* Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)
 |-  ( E! x  x  e.  A  <->  E. x  A  =  { x } )
 
Theoremrabsnt 3678* 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 3679 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  =  { B ,  A }
 
Theorempreq1 3680 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
 |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C } )
 
Theorempreq2 3681 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B } )
 
Theorempreq12 3682 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D }
 )
 
Theorempreq1i 3683 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { A ,  C }  =  { B ,  C }
 
Theorempreq2i 3684 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { C ,  A }  =  { C ,  B }
 
Theorempreq12i 3685 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 { A ,  C }  =  { B ,  D }
 
Theorempreq1d 3686 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  C }
 )
 
Theorempreq2d 3687 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A }  =  { C ,  B }
 )
 
Theorempreq12d 3688 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
 
Theoremtpeq1 3689 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2 3690 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3 3691 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq1d 3692 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2d 3693 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3d 3694 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq123d 3695 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 3696 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 { A ,  B ,  C }  =  { B ,  C ,  A }
 
Theoremtpcoma 3697 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { B ,  A ,  C }
 
Theoremtpcomb 3698 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { A ,  C ,  B }
 
Theoremtpass 3699 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 { A ,  B ,  C }  =  ( { A }  u.  { B ,  C }
 )
 
Theoremqdass 3700 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B ,  C }  u.  { D } )
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