HomeHome Metamath Proof Explorer
Theorem List (p. 38 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21459)
  Hilbert Space Explorer  Hilbert Space Explorer
(21460-22982)
  Users' Mathboxes  Users' Mathboxes
(22983-31404)
 

Theorem List for Metamath Proof Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremqdassr 3701 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 3702 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 3703 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 3704 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 3705 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 }
 
Theoremprid1g 3706 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 3707 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 3708 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 3709 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 3710 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A  e.  _V 
 ->  { A ,  B }  =  { B } )
 
Theoremprprc2 3711 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
 |-  ( -.  B  e.  _V 
 ->  { A ,  B }  =  { A } )
 
Theoremprprc 3712 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 3713 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 3714 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 3715 Closed theorem form of tpid3 3716. This proof was automatically generated from the virtual deduction proof tpid3gVD 27751 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremtpid3 3716 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 3717 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
 |-  ( A  e.  V  ->  { A }  =/=  (/) )
 
Theoremsnnz 3718 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A }  =/= 
 (/)
 
Theoremprnz 3719 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B }  =/=  (/)
 
Theoremprnzg 3720 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  V  ->  { A ,  B }  =/=  (/) )
 
Theoremtpnz 3721 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
 |-  A  e.  _V   =>    |-  { A ,  B ,  C }  =/= 
 (/)
 
Theoremsnss 3722 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 3723 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
 |-  ( A  e.  ( B  \  { C }
 ) 
 <->  ( A  e.  B  /\  A  =/=  C ) )
 
Theoremeldifsni 3724 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
 |-  ( A  e.  ( B  \  { C }
 )  ->  A  =/=  C )
 
Theoremneldifsn 3725  A is not in  ( B 
\  { A }
). (Contributed by David Moews, 1-May-2017.)
 |- 
 -.  A  e.  ( B  \  { A }
 )
 
Theoremneldifsnd 3726  A is not in  ( B 
\  { A }
). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  -.  A  e.  ( B  \  { A } ) )
 
Theoremrexdifsn 3727 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 3728 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 3729 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 )
 
Theoremdifprsn 3730 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 }
 
Theoremdifsneq 3731  ( B  \  { A } ) equals  B if and only if 
A is not a member of  B. Generalization of difsn 3729. (Contributed by David Moews, 1-May-2017.)
 |-  ( -.  A  e.  B 
 <->  ( B  \  { A } )  =  B )
 
Theoremdifsnpss 3732  ( B  \  { A } ) is a proper subclass of  B if and only if  A is a member of  B. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  <->  ( B  \  { A } )  C.  B )
 
Theoremsnssi 3733 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 3734 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 )
 
Theoremdifsnid 3735 If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.)
 |-  ( B  e.  A  ->  ( ( A  \  { B } )  u. 
 { B } )  =  A )
 
Theorempw0 3736 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 (/)  =  { (/) }
 
Theorempwpw0 3737 Compute the power set of the power set of the empty set. (See pw0 3736 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3795, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
 |- 
 ~P { (/) }  =  { (/) ,  { (/) } }
 
Theoremsnsspr1 3738 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
 |- 
 { A }  C_  { A ,  B }
 
Theoremsnsspr2 3739 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
 |- 
 { B }  C_  { A ,  B }
 
Theoremsnsstp1 3740 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { A }  C_  { A ,  B ,  C }
 
Theoremsnsstp2 3741 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { B }  C_  { A ,  B ,  C }
 
Theoremsnsstp3 3742 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { C }  C_  { A ,  B ,  C }
 
Theoremprss 3743 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 3744 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 3745 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 )
 
Theoremsssn 3746 The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  C_  { B } 
 <->  ( A  =  (/)  \/  A  =  { B } ) )
 
Theoremssunsn2 3747 The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 3798. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( ( B  C_  A  /\  A  C_  ( C  u.  { D }
 ) )  <->  ( ( B 
 C_  A  /\  A  C_  C )  \/  (
 ( B  u.  { D } )  C_  A  /\  A  C_  ( C  u.  { D } )
 ) ) )
 
Theoremssunsn 3748 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( ( B  C_  A  /\  A  C_  ( B  u.  { C }
 ) )  <->  ( A  =  B  \/  A  =  ( B  u.  { C } ) ) )
 
Theoremeqsn 3749* Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.)
 |-  ( A  =/=  (/)  ->  ( A  =  { B } 
 <-> 
 A. x  e.  A  x  =  B )
 )
 
Theoremssunpr 3750 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( ( B  C_  A  /\  A  C_  ( B  u.  { C ,  D } ) )  <->  ( ( A  =  B  \/  A  =  ( B  u.  { C } ) )  \/  ( A  =  ( B  u.  { D } )  \/  A  =  ( B  u.  { C ,  D }
 ) ) ) )
 
Theoremsspr 3751 The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)
 |-  ( A  C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
 
Theoremsstp 3752 The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( A  C_  { B ,  C ,  D }  <->  ( ( ( A  =  (/) 
 \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
 ) )  \/  (
 ( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) ) )
 
Theoremtpss 3753 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 )
 
Theoremsneqr 3754 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 3755 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 3756 Closed form of sneqr 3754. (Contributed by Scott Fenton, 1-Apr-2011.)
 |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
 
Theoremsneqbg 3757 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 3758 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A }  C_  ~P A
 
Theoremprsspw 3759 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 ) )
 
Theorempreqr1 3760 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 3761 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 3762 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 3763 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 3764 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 3765 Closed form of preq12b 3762. (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 ) ) ) )
 
Theorempreqsn 3766 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( { A ,  B }  =  { C }  <->  ( A  =  B  /\  B  =  C ) )
 
Theoremdfopif 3767 Rewrite df-op 3623 using  if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
 
Theoremdfopg 3768 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  <. A ,  B >.  =  { { A } ,  { A ,  B } } )
 
Theoremdfop 3769 Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 <. A ,  B >.  =  { { A } ,  { A ,  B } }
 
Theoremopeq1 3770 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  =  B  -> 
 <. A ,  C >.  = 
 <. B ,  C >. )
 
Theoremopeq2 3771 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  A >.  = 
 <. C ,  B >. )
 
Theoremopeq12 3772 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  <. A ,  B >.  =  <. C ,  D >. )
 
Theoremopeq1i 3773 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. A ,  C >.  =  <. B ,  C >.
 
Theoremopeq2i 3774 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. C ,  A >.  =  <. C ,  B >.
 
Theoremopeq12i 3775 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |- 
 <. A ,  C >.  = 
 <. B ,  D >.
 
Theoremopeq1d 3776 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C >.  =  <. B ,  C >. )
 
Theoremopeq2d 3777 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A >.  =  <. C ,  B >. )
 
Theoremopeq12d 3778 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  <. A ,  C >.  = 
 <. B ,  D >. )
 
Theoremoteq1 3779 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. A ,  C ,  D >.  =  <. B ,  C ,  D >. )
 
Theoremoteq2 3780 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  A ,  D >.  =  <. C ,  B ,  D >. )
 
Theoremoteq3 3781 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
 
Theoremoteq1d 3782 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C ,  D >.  = 
 <. B ,  C ,  D >. )
 
Theoremoteq2d 3783 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A ,  D >.  = 
 <. C ,  B ,  D >. )
 
Theoremoteq3d 3784 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  D ,  A >.  = 
 <. C ,  D ,  B >. )
 
Theoremoteq123d 3785 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   &    |-  ( ph  ->  E  =  F )   =>    |-  ( ph  ->  <. A ,  C ,  E >.  = 
 <. B ,  D ,  F >. )
 
Theoremnfop 3786 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x <. A ,  B >.
 
Theoremnfopd 3787 Deduction version of bound-variable hypothesis builder nfop 3786. This shows how the deduction version of a not-free theorem such as nfop 3786 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x <. A ,  B >. )
 
Theoremopid 3788 The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 |-  A  e.  _V   =>    |-  <. A ,  A >.  =  { { A } }
 
Theoremralunsn 3789* Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   =>    |-  ( B  e.  C  ->  ( A. x  e.  ( A  u.  { B } ) ph  <->  ( A. x  e.  A  ph  /\  ps )
 ) )
 
Theorem2ralunsn 3790* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( x  =  B  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  B  ->  ( ps  <->  th ) )   =>    |-  ( B  e.  C  ->  ( A. x  e.  ( A  u.  { B } ) A. y  e.  ( A  u.  { B } ) ph  <->  ( ( A. x  e.  A  A. y  e.  A  ph  /\  A. x  e.  A  ps )  /\  ( A. y  e.  A  ch  /\  th ) ) ) )
 
Theoremopprc 3791 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  <. A ,  B >.  =  (/) )
 
Theoremopprc1 3792 Expansion of an ordered pair when the first member is a proper class. See also opprc 3791. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  A  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremopprc2 3793 Expansion of an ordered pair when the second member is a proper class. See also opprc 3791. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  B  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremoprcl 3794 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( C  e.  <. A ,  B >.  ->  ( A  e.  _V  /\  B  e.  _V ) )
 
Theorempwsn 3795 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
TheorempwsnALT 3796 The power set of a singleton (direct proof). (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
Theorempwpr 3797 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
 |- 
 ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
 
Theorempwtp 3798 The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |- 
 ~P { A ,  B ,  C }  =  ( ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )  u.  ( { { C } ,  { A ,  C } }  u.  { { B ,  C } ,  { A ,  B ,  C } } ) )
 
Theorempwpwpw0 3799 Compute the power set of the power set of the power set of the empty set. (See also pw0 3736 and pwpw0 3737.) (Contributed by NM, 2-May-2009.)
 |- 
 ~P { (/) ,  { (/)
 } }  =  ( { (/) ,  { (/) } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } } )
 
Theorempwv 3800 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
 |- 
 ~P _V  =  _V
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31404
  Copyright terms: Public domain < Previous  Next >