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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdifsnpss 3901  ( 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 3902 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 3903 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 3904 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 3905 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 3906 Compute the power set of the power set of the empty set. (See pw0 3905 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3969, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
 |- 
 ~P { (/) }  =  { (/) ,  { (/) } }
 
Theoremsnsspr1 3907 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
 |- 
 { A }  C_  { A ,  B }
 
Theoremsnsspr2 3908 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
 |- 
 { B }  C_  { A ,  B }
 
Theoremsnsstp1 3909 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { A }  C_  { A ,  B ,  C }
 
Theoremsnsstp2 3910 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { B }  C_  { A ,  B ,  C }
 
Theoremsnsstp3 3911 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
 |- 
 { C }  C_  { A ,  B ,  C }
 
Theoremprss 3912 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 3913 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 3914 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 3915 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 ) ) )
 
TheoremprsspwgOLD 3916 Obsolete version of prsspwg 3915 as of 18-Jan-2018. (Contributed by Thierry Arnoux, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )
 
Theoremsssn 3917 The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  C_  { B } 
 <->  ( A  =  (/)  \/  A  =  { B } ) )
 
Theoremssunsn2 3918 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 3972. (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 3919 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 3920* 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 3921 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 3922 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 3923 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 3924 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 3925 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 3926 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 3927 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 3928 Closed form of sneqr 3926. (Contributed by Scott Fenton, 1-Apr-2011.)
 |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
 
Theoremsneqbg 3929 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 3930 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
 |- 
 { A }  C_  ~P A
 
Theoremprsspw 3931 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 3932 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 3933 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 3934 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 3935 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 3936 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 3937 Closed form of preq12b 3934. (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 ) ) ) )
 
Theoremprneimg 3938 Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( ( ( A  e.  U  /\  B  e.  V )  /\  ( C  e.  X  /\  D  e.  Y )
 )  ->  ( (
 ( A  =/=  C  /\  A  =/=  D )  \/  ( B  =/=  C 
 /\  B  =/=  D ) )  ->  { A ,  B }  =/=  { C ,  D }
 ) )
 
Theoremprnebg 3939 A (proper) pair is not equal to another (maybe inproper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.)
 |-  ( ( ( A  e.  U  /\  B  e.  V )  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B ) 
 ->  ( ( ( A  =/=  C  /\  A  =/=  D )  \/  ( B  =/=  C  /\  B  =/=  D ) )  <->  { A ,  B }  =/=  { C ,  D } ) )
 
Theorempreqsn 3940 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 3941 Rewrite df-op 3783 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 3942 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 3943 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 3944 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 3945 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 3946 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  <. A ,  B >.  =  <. C ,  D >. )
 
Theoremopeq1i 3947 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. A ,  C >.  =  <. B ,  C >.
 
Theoremopeq2i 3948 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. C ,  A >.  =  <. C ,  B >.
 
Theoremopeq12i 3949 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 3950 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C >.  =  <. B ,  C >. )
 
Theoremopeq2d 3951 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A >.  =  <. C ,  B >. )
 
Theoremopeq12d 3952 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 3953 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. A ,  C ,  D >.  =  <. B ,  C ,  D >. )
 
Theoremoteq2 3954 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  A ,  D >.  =  <. C ,  B ,  D >. )
 
Theoremoteq3 3955 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
 
Theoremoteq1d 3956 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C ,  D >.  = 
 <. B ,  C ,  D >. )
 
Theoremoteq2d 3957 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A ,  D >.  = 
 <. C ,  B ,  D >. )
 
Theoremoteq3d 3958 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  D ,  A >.  = 
 <. C ,  D ,  B >. )
 
Theoremoteq123d 3959 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 3960 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x <. A ,  B >.
 
Theoremnfopd 3961 Deduction version of bound-variable hypothesis builder nfop 3960. This shows how the deduction version of a not-free theorem such as nfop 3960 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 3962 The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 |-  A  e.  _V   =>    |-  <. A ,  A >.  =  { { A } }
 
Theoremralunsn 3963* 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 3964* 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 3965 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 3966 Expansion of an ordered pair when the first member is a proper class. See also opprc 3965. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  A  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremopprc2 3967 Expansion of an ordered pair when the second member is a proper class. See also opprc 3965. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  B  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremoprcl 3968 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 3969 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
TheorempwsnALT 3970 The power set of a singleton (direct proof). TO DO - should we keep this? (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
Theorempwpr 3971 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
 |- 
 ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
 
Theorempwtp 3972 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 3973 Compute the power set of the power set of the power set of the empty set. (See also pw0 3905 and pwpw0 3906.) (Contributed by NM, 2-May-2009.)
 |- 
 ~P { (/) ,  { (/)
 } }  =  ( { (/) ,  { (/) } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } } )
 
Theorempwv 3974 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
 
2.1.18  The union of a class
 
Syntaxcuni 3975 Extend class notation to include the union of a class (read: 'union  A')
 class  U. A
 
Definitiondf-uni 3976* Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example,  U. { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
1 ,  3 ,  8 } (ex-uni 21687). This is similar to the union of two classes df-un 3285. (Contributed by NM, 23-Aug-1993.)
 |- 
 U. A  =  { x  |  E. y
 ( x  e.  y  /\  y  e.  A ) }
 
Theoremdfuni2 3977* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
 |- 
 U. A  =  { x  |  E. y  e.  A  x  e.  y }
 
Theoremeluni 3978* Membership in class union. (Contributed by NM, 22-May-1994.)
 |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremeluni2 3979* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
 |-  ( A  e.  U. B 
 <-> 
 E. x  e.  B  A  e.  x )
 
Theoremelunii 3980 Membership in class union. (Contributed by NM, 24-Mar-1995.)
 |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C )
 
Theoremnfuni 3981 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  F/_ x A   =>    |-  F/_ x U. A
 
Theoremnfunid 3982 Deduction version of nfuni 3981. (Contributed by NM, 18-Feb-2013.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x U. A )
 
Theoremcsbunig 3983 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 U. B  =  U. [_ A  /  x ]_ B )
 
Theoremunieq 3984 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  =  B  ->  U. A  =  U. B )
 
Theoremunieqi 3985 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  U. A  =  U. B
 
Theoremunieqd 3986 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  U. A  =  U. B )
 
Theoremeluniab 3987* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  e.  U. { x  |  ph }  <->  E. x ( A  e.  x  /\  ph )
 )
 
Theoremelunirab 3988* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
 |-  ( A  e.  U. { x  e.  B  |  ph
 } 
 <-> 
 E. x  e.  B  ( A  e.  x  /\  ph ) )
 
Theoremunipr 3989 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 U. { A ,  B }  =  ( A  u.  B )
 
Theoremuniprg 3990 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. { A ,  B }  =  ( A  u.  B ) )
 
Theoremunisn 3991 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  U. { A }  =  A
 
Theoremunisng 3992 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
 |-  ( A  e.  V  ->  U. { A }  =  A )
 
Theoremdfnfc2 3993* An alternative statement of the effective freeness of a class  A, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( A. x  A  e.  V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A )
 )
 
Theoremuniun 3994 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
 |- 
 U. ( A  u.  B )  =  ( U. A  u.  U. B )
 
Theoremuniin 3995 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 6943 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. ( A  i^i  B )  C_  ( U. A  i^i  U. B )
 
Theoremuniss 3996 Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  C_  B  ->  U. A  C_  U. B )
 
Theoremssuni 3997 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )
 
Theoremunissi 3998 Subclass relationship for subclass union. Inference form of uniss 3996. (Contributed by David Moews, 1-May-2017.)
 |-  A  C_  B   =>    |- 
 U. A  C_  U. B
 
Theoremunissd 3999 Subclass relationship for subclass union. Deduction form of uniss 3996. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  U. A  C_ 
 U. B )
 
Theoremuni0b 4000 The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
 |-  ( U. A  =  (/)  <->  A 
 C_  { (/) } )
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