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Theorem List for Metamath Proof Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprsspw 3801 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 3802 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 3803 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 3804 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 3805 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 3806 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 3807 Closed form of preq12b 3804. (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 3808 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 3809 Rewrite df-op 3662 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 3810 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 3811 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 3812 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 3813 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 3814 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  <. A ,  B >.  =  <. C ,  D >. )
 
Theoremopeq1i 3815 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. A ,  C >.  =  <. B ,  C >.
 
Theoremopeq2i 3816 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. C ,  A >.  =  <. C ,  B >.
 
Theoremopeq12i 3817 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 3818 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C >.  =  <. B ,  C >. )
 
Theoremopeq2d 3819 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A >.  =  <. C ,  B >. )
 
Theoremopeq12d 3820 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 3821 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. A ,  C ,  D >.  =  <. B ,  C ,  D >. )
 
Theoremoteq2 3822 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  A ,  D >.  =  <. C ,  B ,  D >. )
 
Theoremoteq3 3823 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
 
Theoremoteq1d 3824 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C ,  D >.  = 
 <. B ,  C ,  D >. )
 
Theoremoteq2d 3825 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A ,  D >.  = 
 <. C ,  B ,  D >. )
 
Theoremoteq3d 3826 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  D ,  A >.  = 
 <. C ,  D ,  B >. )
 
Theoremoteq123d 3827 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 3828 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x <. A ,  B >.
 
Theoremnfopd 3829 Deduction version of bound-variable hypothesis builder nfop 3828. This shows how the deduction version of a not-free theorem such as nfop 3828 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 3830 The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 |-  A  e.  _V   =>    |-  <. A ,  A >.  =  { { A } }
 
Theoremralunsn 3831* 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 3832* 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 3833 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 3834 Expansion of an ordered pair when the first member is a proper class. See also opprc 3833. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  A  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremopprc2 3835 Expansion of an ordered pair when the second member is a proper class. See also opprc 3833. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  B  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremoprcl 3836 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 3837 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
TheorempwsnALT 3838 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 3839 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
 |- 
 ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
 
Theorempwtp 3840 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 3841 Compute the power set of the power set of the power set of the empty set. (See also pw0 3778 and pwpw0 3779.) (Contributed by NM, 2-May-2009.)
 |- 
 ~P { (/) ,  { (/)
 } }  =  ( { (/) ,  { (/) } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } } )
 
Theorempwv 3842 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 3843 Extend class notation to include the union of a class (read: 'union  A')
 class  U. A
 
Definitiondf-uni 3844* 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 20829). This is similar to the union of two classes df-un 3170. (Contributed by NM, 23-Aug-1993.)
 |- 
 U. A  =  { x  |  E. y
 ( x  e.  y  /\  y  e.  A ) }
 
Theoremdfuni2 3845* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
 |- 
 U. A  =  { x  |  E. y  e.  A  x  e.  y }
 
Theoremeluni 3846* Membership in class union. (Contributed by NM, 22-May-1994.)
 |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremeluni2 3847* 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 3848 Membership in class union. (Contributed by NM, 24-Mar-1995.)
 |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C )
 
Theoremnfuni 3849 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 3850 Deduction version of nfuni 3849. (Contributed by NM, 18-Feb-2013.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x U. A )
 
Theoremcsbunig 3851 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 3852 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 3853 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  U. A  =  U. B
 
Theoremunieqd 3854 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  U. A  =  U. B )
 
Theoremeluniab 3855* 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 3856* 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 3857 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 3858 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 3859 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 3860 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 3861* 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 3862 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 3863 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uninqs 25142 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 3864 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 3865 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 3866 Subclass relationship for subclass union. Inference form of uniss 3864. (Contributed by David Moews, 1-May-2017.)
 |-  A  C_  B   =>    |- 
 U. A  C_  U. B
 
Theoremunissd 3867 Subclass relationship for subclass union. Deduction form of uniss 3864. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  U. A  C_ 
 U. B )
 
Theoremuni0b 3868 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_  { (/) } )
 
Theoremuni0c 3869* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
 |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
 
Theoremuni0 3870 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4165 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
 |- 
 U. (/)  =  (/)
 
Theoremelssuni 3871 An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)
 |-  ( A  e.  B  ->  A  C_  U. B )
 
Theoremunissel 3872 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
 |-  ( ( U. A  C_  B  /\  B  e.  A )  ->  U. A  =  B )
 
Theoremunissb 3873* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
 |-  ( U. A  C_  B 
 <-> 
 A. x  e.  A  x  C_  B )
 
Theoremuniss2 3874* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 3963 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
 |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  U. A  C_  U. B )
 
Theoremunidif 3875* If the difference  A  \  B contains the largest members of  A, then the union of the difference is the union of  A. (Contributed by NM, 22-Mar-2004.)
 |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
 
Theoremssunieq 3876* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  x  C_  A )  ->  A  =  U. B )
 
Theoremunimax 3877* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
 |-  ( A  e.  B  ->  U. { x  e.  B  |  x  C_  A }  =  A )
 
2.1.19  The intersection of a class
 
Syntaxcint 3878 Extend class notation to include the intersection of a class (read: 'intersect  A').
 class  |^| A
 
Definitiondf-int 3879* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example,  |^| { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
1 }. Compare this with the intersection of two classes, df-in 3172. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
 
Theoremdfint2 3880* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  { x  |  A. y  e.  A  x  e.  y }
 
Theoreminteq 3881 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
 |-  ( A  =  B  -> 
 |^| A  =  |^| B )
 
Theoreminteqi 3882 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  A  =  B   =>    |-  |^| A  =  |^| B
 
Theoreminteqd 3883 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  |^| A  =  |^| B )
 
Theoremelint 3884* Membership in class intersection. (Contributed by NM, 21-May-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x ( x  e.  B  ->  A  e.  x ) )
 
Theoremelint2 3885* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x  e.  B  A  e.  x )
 
Theoremelintg 3886* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
 |-  ( A  e.  V  ->  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
 )
 
Theoremelinti 3887 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  e.  |^| B 
 ->  ( C  e.  B  ->  A  e.  C ) )
 
Theoremnfint 3888 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  F/_ x A   =>    |-  F/_ x |^| A
 
Theoremelintab 3889* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 { x  |  ph }  <->  A. x ( ph  ->  A  e.  x ) )
 
Theoremelintrab 3890* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 { x  e.  B  |  ph }  <->  A. x  e.  B  ( ph  ->  A  e.  x ) )
 
Theoremelintrabg 3891* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  B  |  ph
 } 
 <-> 
 A. x  e.  B  ( ph  ->  A  e.  x ) ) )
 
Theoremint0 3892 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| (/)  =  _V
 
Theoremintss1 3893 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
 |-  ( A  e.  B  -> 
 |^| B  C_  A )
 
Theoremssint 3894* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  |^| B  <->  A. x  e.  B  A  C_  x )
 
Theoremssintab 3895* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  C_  |^| { x  |  ph }  <->  A. x ( ph  ->  A  C_  x )
 )
 
Theoremssintub 3896* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
 |-  A  C_  |^| { x  e.  B  |  A  C_  x }
 
Theoremssmin 3897* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
 |-  A  C_  |^| { x  |  ( A  C_  x  /\  ph ) }
 
Theoremintmin 3898* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( A  e.  B  -> 
 |^| { x  e.  B  |  A  C_  x }  =  A )
 
Theoremintss 3899 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  -> 
 |^| B  C_  |^| A )
 
Theoremintssuni 3900 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
 |-  ( A  =/=  (/)  ->  |^| A  C_ 
 U. A )
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