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Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprcom 3501 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A ,  B }  =  { B ,  A }
 
Theorempreq1 3502 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
 |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C } )
 
Theorempreq2 3503 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B } )
 
Theorempreq12 3504 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D }
 )
 
Theorempreq1i 3505 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { A ,  C }  =  { B ,  C }
 
Theorempreq2i 3506 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   =>    |-  { C ,  A }  =  { C ,  B }
 
Theorempreq12i 3507 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 { A ,  C }  =  { B ,  D }
 
Theorempreq1d 3508 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  C }
 )
 
Theorempreq2d 3509 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A }  =  { C ,  B }
 )
 
Theorempreq12d 3510 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
 
Theoremtpeq1 3511 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2 3512 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3 3513 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
 |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq1d 3514 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { A ,  C ,  D }  =  { B ,  C ,  D } )
 
Theoremtpeq2d 3515 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  A ,  D }  =  { C ,  B ,  D } )
 
Theoremtpeq3d 3516 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
 
Theoremtpeq123d 3517 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 3518 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
 |- 
 { A ,  B ,  C }  =  { B ,  C ,  A }
 
Theoremtpcoma 3519 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { B ,  A ,  C }
 
Theoremtpcomb 3520 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
 |- 
 { A ,  B ,  C }  =  { A ,  C ,  B }
 
Theoremtpass 3521 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 { A ,  B ,  C }  =  ( { A }  u.  { B ,  C }
 )
 
Theoremqdass 3522 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 3523 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 3524 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 3525 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 3526 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 3527 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 3528 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 3529 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 3530 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 3531 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 3532 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 3533 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  e.  _V 
 ->  { A ,  B }  =  { B } )
 
Theoremprprc2 3534 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
 |-  ( -.  B  e.  _V 
 ->  { A ,  B }  =  { A } )
 
Theoremprprc 3535 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 3536 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 3537 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 3538 Closed theorem form of tpid3 3539. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremtpid3 3539 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 3540 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  e.  V  ->  { A }  =/=  (/) )
 
Theoremsnmg 3541* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  ( A  e.  V  ->  E. x  x  e. 
 { A } )
 
Theoremsnnz 3542 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A }  =/= 
 (/)
 
Theoremsnm 3543* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
 |-  A  e.  _V   =>    |-  E. x  x  e.  { A }
 
Theoremprmg 3544* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( A  e.  V  ->  E. x  x  e. 
 { A ,  B } )
 
Theoremprnz 3545 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B }  =/=  (/)
 
Theoremprm 3546* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  A  e.  _V   =>    |-  E. x  x  e.  { A ,  B }
 
Theoremprnzg 3547 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
 
Theoremtpnz 3548 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B ,  C }  =/= 
 (/)
 
Theoremsnss 3549 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 3550 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
 |-  ( A  e.  ( B  \  { C }
 ) 
 <->  ( A  e.  B  /\  A  =/=  C ) )
 
Theoremssdifsn 3551 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 3552 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
 |-  ( A  e.  ( B  \  { C }
 )  ->  A  =/=  C )
 
Theoremneldifsn 3553  A is not in  ( B 
\  { A }
). (Contributed by David Moews, 1-May-2017.)
 |- 
 -.  A  e.  ( B  \  { A }
 )
 
Theoremneldifsnd 3554  A is not in  ( B 
\  { A }
). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  A  e.  ( B  \  { A } ) )
 
Theoremrexdifsn 3555 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 3556 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 3557 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 3558 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 3559 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B } )
 
Theoremdifprsn2 3560 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A } )
 
Theoremdiftpsn3 3561 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 3562 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 3563  ( B  \  { A } ) equals  B if and only if 
A is not a member of  B. Generalization of difsn 3557. (Contributed by David Moews, 1-May-2017.)
 |-  ( -.  A  e.  B 
 <->  ( B  \  { A } )  =  B )
 
Theoremsnssi 3564 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 3565 The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  { A }  C_  B )
 
Theoremdifsnss 3566 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 6218. (Contributed by Jim Kingdon, 10-Aug-2018.)
 |-  ( B  e.  A  ->  ( ( A  \  { B } )  u. 
 { B } )  C_  A )
 
Theorempw0 3567 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 3568 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
 |- 
 { A }  C_  { A ,  B }
 
Theoremsnsspr2 3569 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
 |- 
 { B }  C_  { A ,  B }
 
Theoremsnsstp1 3570 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { A }  C_  { A ,  B ,  C }
 
Theoremsnsstp2 3571 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { B }  C_  { A ,  B ,  C }
 
Theoremsnsstp3 3572 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { C }  C_  { A ,  B ,  C }
 
Theoremprsstp12 3573 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 3574 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 3575 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 3576 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 3577 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 3578 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 3579 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 3580 Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
 |-  ( ( A  =  (/) 
 \/  A  =  { B } )  ->  A  C_ 
 { B } )
 
Theoremsssnm 3581* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  ( A  C_  { B }  <->  A  =  { B }
 ) )
 
Theoremeqsnm 3582* 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 3583 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 3584 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 3585 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 3586 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 3587 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 3588 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 3589 Closed form of sneqr 3587. (Contributed by Scott Fenton, 1-Apr-2011.)
 |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
 
Theoremsneqbg 3590 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 3591 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A }  C_  ~P A
 
Theoremprsspw 3592 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 3593 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3595. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
 
Theorempreqr2g 3594 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3596. (Contributed by Jim Kingdon, 21-Sep-2018.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V )  ->  ( { C ,  A }  =  { C ,  B }  ->  A  =  B ) )
 
Theorempreqr1 3595 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
 
Theorempreqr2 3596 Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { C ,  A }  =  { C ,  B }  ->  A  =  B )
 
Theorempreq12b 3597 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { A ,  B }  =  { C ,  D }  <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) )
 
Theoremprel12 3598 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( -.  A  =  B  ->  ( { A ,  B }  =  { C ,  D }  <->  ( A  e.  { C ,  D }  /\  B  e.  { C ,  D } ) ) )
 
Theoremopthpr 3599 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  D  ->  ( { A ,  B }  =  { C ,  D }  <->  ( A  =  C  /\  B  =  D )
 ) )
 
Theorempreq12bg 3600 Closed form of preq12b 3597. (Contributed by Scott Fenton, 28-Mar-2014.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
 )  ->  ( { A ,  B }  =  { C ,  D } 
 <->  ( ( A  =  C  /\  B  =  D )  \/  ( A  =  D  /\  B  =  C ) ) ) )
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