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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreusn 3601* 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 3602 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
 |-  ( ( A  e.  V  /\  { x  |  ph
 }  =  { A } )  ->  E! x ph )
 
Theoremrabsneu 3603 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 3604* Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)
 |-  ( E! x  x  e.  A  <->  E. x  A  =  { x } )
 
Theoremrabsnt 3605* 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 3606 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  =  { B ,  A }
 
Theorempreq1 3607 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
 |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C } )
 
Theorempreq2 3608 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B } )
 
Theorempreq12 3609 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D }
 )
 
Theorempreq1i 3610 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { A ,  C }  =  { B ,  C }
 
Theorempreq2i 3611 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { C ,  A }  =  { C ,  B }
 
Theorempreq12i 3612 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 { A ,  C }  =  { B ,  D }
 
Theorempreq1d 3613 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  C }
 )
 
Theorempreq2d 3614 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A }  =  { C ,  B }
 )
 
Theorempreq12d 3615 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
 
Theoremtpeq1 3616 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2 3617 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3 3618 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq1d 3619 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2d 3620 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3d 3621 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq123d 3622 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 3623 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 { A ,  B ,  C }  =  { B ,  C ,  A }
 
Theoremtpcoma 3624 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { B ,  A ,  C }
 
Theoremtpcomb 3625 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { A ,  C ,  B }
 
Theoremtpass 3626 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 { A ,  B ,  C }  =  ( { A }  u.  { B ,  C }
 )
 
Theoremqdass 3627 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 3628 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 3629 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 3630 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 3631 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 3632 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 3633 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 3634 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 3635 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 3636 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  A  e.  { A ,  B }
 
Theoremprid2 3637 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 3638 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  e.  _V 
 ->  { A ,  B }  =  { B } )
 
Theoremprprc2 3639 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
 |-  ( -.  B  e.  _V 
 ->  { A ,  B }  =  { A } )
 
Theoremprprc 3640 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 3641 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 }
 
Theoremtpid1g 3642 Closed theorem form of tpid1 3641. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
 |-  ( A  e.  B  ->  A  e.  { A ,  C ,  D }
 )
 
Theoremtpid2 3643 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 }
 
Theoremtpid2g 3644 Closed theorem form of tpid2 3643. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
 |-  ( A  e.  B  ->  A  e.  { C ,  A ,  D }
 )
 
Theoremtpid3g 3645 Closed theorem form of tpid3 3646. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremtpid3 3646 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 3647 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  e.  V  ->  { A }  =/=  (/) )
 
Theoremsnmg 3648* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  ( A  e.  V  ->  E. x  x  e. 
 { A } )
 
Theoremsnnz 3649 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A }  =/= 
 (/)
 
Theoremsnm 3650* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  A  e.  _V   =>    |-  E. x  x  e.  { A }
 
Theoremprmg 3651* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( A  e.  V  ->  E. x  x  e. 
 { A ,  B } )
 
Theoremprnz 3652 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B }  =/=  (/)
 
Theoremprm 3653* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  A  e.  _V   =>    |-  E. x  x  e.  { A ,  B }
 
Theoremprnzg 3654 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
 
Theoremtpnz 3655 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B ,  C }  =/= 
 (/)
 
Theoremsnss 3656 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <->  { A }  C_  B )
 
Theoremeldifsn 3657 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
 |-  ( A  e.  ( B  \  { C }
 ) 
 <->  ( A  e.  B  /\  A  =/=  C ) )
 
Theoremssdifsn 3658 Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)
 |-  ( A  C_  ( B  \  { C }
 ) 
 <->  ( A  C_  B  /\  -.  C  e.  A ) )
 
Theoremeldifsni 3659 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
 |-  ( A  e.  ( B  \  { C }
 )  ->  A  =/=  C )
 
Theoremneldifsn 3660  A is not in  ( B 
\  { A }
). (Contributed by David Moews, 1-May-2017.)
 |- 
 -.  A  e.  ( B  \  { A }
 )
 
Theoremneldifsnd 3661  A is not in  ( B 
\  { A }
). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  A  e.  ( B  \  { A } ) )
 
Theoremrexdifsn 3662 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
 |-  ( E. x  e.  ( A  \  { B } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  ph ) )
 
Theoremsnssg 3663 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
 |-  ( A  e.  V  ->  ( A  e.  B  <->  { A }  C_  B ) )
 
Theoremdifsn 3664 An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( -.  A  e.  B  ->  ( B  \  { A } )  =  B )
 
Theoremdifprsnss 3665 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( { A ,  B }  \  { A } )  C_  { B }
 
Theoremdifprsn1 3666 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B } )
 
Theoremdifprsn2 3667 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A } )
 
Theoremdiftpsn3 3668 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( ( A  =/=  C 
 /\  B  =/=  C )  ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
 
Theoremdifpr 3669 Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
 |-  ( A  \  { B ,  C }
 )  =  ( ( A  \  { B } )  \  { C } )
 
Theoremdifsnb 3670  ( B  \  { A } ) equals  B if and only if 
A is not a member of  B. Generalization of difsn 3664. (Contributed by David Moews, 1-May-2017.)
 |-  ( -.  A  e.  B 
 <->  ( B  \  { A } )  =  B )
 
Theoremsnssi 3671 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
 |-  ( A  e.  B  ->  { A }  C_  B )
 
Theoremsnssd 3672 The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  { A }  C_  B )
 
Theoremdifsnss 3673 If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6410. (Contributed by Jim Kingdon, 10-Aug-2018.)
 |-  ( B  e.  A  ->  ( ( A  \  { B } )  u. 
 { B } )  C_  A )
 
Theorempw0 3674 Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 ~P (/)  =  { (/) }
 
Theoremsnsspr1 3675 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
 |- 
 { A }  C_  { A ,  B }
 
Theoremsnsspr2 3676 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
 |- 
 { B }  C_  { A ,  B }
 
Theoremsnsstp1 3677 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { A }  C_  { A ,  B ,  C }
 
Theoremsnsstp2 3678 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { B }  C_  { A ,  B ,  C }
 
Theoremsnsstp3 3679 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { C }  C_  { A ,  B ,  C }
 
Theoremprsstp12 3680 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |- 
 { A ,  B }  C_  { A ,  B ,  C }
 
Theoremprsstp13 3681 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |- 
 { A ,  C }  C_  { A ,  B ,  C }
 
Theoremprsstp23 3682 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |- 
 { B ,  C }  C_  { A ,  B ,  C }
 
Theoremprss 3683 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  C  /\  B  e.  C ) 
 <->  { A ,  B }  C_  C )
 
Theoremprssg 3684 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C ) )
 
Theoremprssi 3685 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
 |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
 
Theoremprsspwg 3686 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )
 
Theoremsssnr 3687 Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4132. (Contributed by Jim Kingdon, 10-Aug-2018.)
 |-  ( ( A  =  (/) 
 \/  A  =  { B } )  ->  A  C_ 
 { B } )
 
Theoremsssnm 3688* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  ( A  C_  { B }  <->  A  =  { B }
 ) )
 
Theoremeqsnm 3689* Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  ( A  =  { B } 
 <-> 
 A. x  e.  A  x  =  B )
 )
 
Theoremssprr 3690 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) )  ->  A  C_  { B ,  C } )
 
Theoremsstpr 3691 The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  ( ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) )  \/  ( ( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) )  ->  A  C_  { B ,  C ,  D }
 )
 
Theoremtpss 3692 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D )
 
Theoremtpssi 3693 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D ) 
 ->  { A ,  B ,  C }  C_  D )
 
Theoremsneqr 3694 If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( { A }  =  { B }  ->  A  =  B )
 
Theoremsnsssn 3695 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
 |-  A  e.  _V   =>    |-  ( { A }  C_  { B }  ->  A  =  B )
 
Theoremsneqrg 3696 Closed form of sneqr 3694. (Contributed by Scott Fenton, 1-Apr-2011.)
 |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
 
Theoremsneqbg 3697 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  ( A  e.  V  ->  ( { A }  =  { B }  <->  A  =  B ) )
 
Theoremsnsspw 3698 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A }  C_  ~P A
 
Theoremprsspw 3699 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) )
 
Theorempreqr1g 3700 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3702. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
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