HomeHome Metamath Proof Explorer
Theorem List (p. 38 of 310)
< 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
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30955)
 

Theorem List for Metamath Proof Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopeq12i 3701 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 3702 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C >.  =  <. B ,  C >. )
 
Theoremopeq2d 3703 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A >.  =  <. C ,  B >. )
 
Theoremopeq12d 3704 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 3705 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. A ,  C ,  D >.  =  <. B ,  C ,  D >. )
 
Theoremoteq2 3706 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  A ,  D >.  =  <. C ,  B ,  D >. )
 
Theoremoteq3 3707 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
 
Theoremoteq1d 3708 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C ,  D >.  = 
 <. B ,  C ,  D >. )
 
Theoremoteq2d 3709 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A ,  D >.  = 
 <. C ,  B ,  D >. )
 
Theoremoteq3d 3710 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  D ,  A >.  = 
 <. C ,  D ,  B >. )
 
Theoremoteq123d 3711 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 3712 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x <. A ,  B >.
 
Theoremnfopd 3713 Deduction version of bound-variable hypothesis builder nfop 3712. This shows how the deduction version of a not-free theorem such as nfop 3712 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 3714 The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 |-  A  e.  _V   =>    |-  <. A ,  A >.  =  { { A } }
 
Theoremralunsn 3715* 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 3716* 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 3717 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 3718 Expansion of an ordered pair when the first member is a proper class. See also opprc 3717. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  A  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremopprc2 3719 Expansion of an ordered pair when the second member is a proper class. See also opprc 3717. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  B  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremoprcl 3720 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 3721 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
TheorempwsnALT 3722 The power set of a singleton (direct proof). (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
Theorempwpr 3723 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
 |- 
 ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
 
Theorempwtp 3724 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 3725 Compute the power set of the power set of the power set of the empty set. (See also pw0 3662 and pwpw0 3663.) (Contributed by NM, 2-May-2009.)
 |- 
 ~P { (/) ,  { (/)
 } }  =  ( { (/) ,  { (/) } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } } )
 
Theorempwv 3726 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 3727 Extend class notation to include the union of a class (read: 'union  A')
 class  U. A
 
Definitiondf-uni 3728* 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 20626). This is similar to the union of two classes df-un 3083. (Contributed by NM, 23-Aug-1993.)
 |- 
 U. A  =  { x  |  E. y
 ( x  e.  y  /\  y  e.  A ) }
 
Theoremdfuni2 3729* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
 |- 
 U. A  =  { x  |  E. y  e.  A  x  e.  y }
 
Theoremeluni 3730* Membership in class union. (Contributed by NM, 22-May-1994.)
 |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremeluni2 3731* 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 3732 Membership in class union. (Contributed by NM, 24-Mar-1995.)
 |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C )
 
Theoremnfuni 3733 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 3734 Deduction version of nfuni 3733. (Contributed by NM, 18-Feb-2013.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x U. A )
 
Theoremcsbunig 3735 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 3736 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 3737 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  U. A  =  U. B
 
Theoremunieqd 3738 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  U. A  =  U. B )
 
Theoremeluniab 3739* 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 3740* 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 3741 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 3742 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 3743 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 3744 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 3745* 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 3746 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 3747 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uninqs 24204 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 3748 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 3749 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 )
 
Theoremuni0b 3750 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 3751* 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 3752 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4046 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
 |- 
 U. (/)  =  (/)
 
Theoremelssuni 3753 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 3754 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 3755* 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 3756* 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 3845 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 3757* 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 3758* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  x  C_  A )  ->  A  =  U. B )
 
Theoremunimax 3759* 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 3760 Extend class notation to include the intersection of a class (read: 'intersect  A').
 class  |^| A
 
Definitiondf-int 3761* 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 3085. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
 
Theoremdfint2 3762* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  { x  |  A. y  e.  A  x  e.  y }
 
Theoreminteq 3763 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
 |-  ( A  =  B  -> 
 |^| A  =  |^| B )
 
Theoreminteqi 3764 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  A  =  B   =>    |-  |^| A  =  |^| B
 
Theoreminteqd 3765 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  |^| A  =  |^| B )
 
Theoremelint 3766* 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 3767* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x  e.  B  A  e.  x )
 
Theoremelintg 3768* 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 3769 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 3770 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 3771* 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 3772* 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 3773* 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 3774 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| (/)  =  _V
 
Theoremintss1 3775 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 3776* 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 3777* 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 3778* Subclass of a least upper bound. (Contributed by NM, 8-Aug-2000.)
 |-  A  C_  |^| { x  e.  B  |  A  C_  x }
 
Theoremssmin 3779* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
 |-  A  C_  |^| { x  |  ( A  C_  x  /\  ph ) }
 
Theoremintmin 3780* 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 3781 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  -> 
 |^| B  C_  |^| A )
 
Theoremintssuni 3782 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
 |-  ( A  =/=  (/)  ->  |^| A  C_ 
 U. A )
 
Theoremssintrab 3783* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
 |-  ( A  C_  |^| { x  e.  B  |  ph }  <->  A. x  e.  B  ( ph  ->  A  C_  x ) )
 
Theoremunissint 3784 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3797). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( U. A  C_  |^|
 A 
 <->  ( A  =  (/)  \/ 
 U. A  =  |^| A ) )
 
Theoremintssuni2 3785 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_  U. B )
 
Theoremintminss 3786* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
 
Theoremintmin2 3787* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
 |-  A  e.  _V   =>    |-  |^| { x  |  A  C_  x }  =  A
 
Theoremintmin3 3788* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  ( A  e.  V  ->  |^|
 { x  |  ph } 
 C_  A )
 
Theoremintmin4 3789* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
 |-  ( A  C_  |^| { x  |  ph }  ->  |^| { x  |  ( A  C_  x  /\  ph ) }  =  |^|
 { x  |  ph } )
 
Theoremintab 3790* The intersection of a special case of a class abstraction.  y may be free in  ph and  A, which can be thought of a  ph ( y ) and  A ( y ). Typically, abrexex2 5632 or abexssex 5633 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
 |-  A  e.  _V   &    |-  { x  |  E. y ( ph  /\  x  =  A ) }  e.  _V   =>    |-  |^| { x  |  A. y ( ph  ->  A  e.  x ) }  =  { x  |  E. y ( ph  /\  x  =  A ) }
 
Theoremint0el 3791 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  e.  A  -> 
 |^| A  =  (/) )
 
Theoremintun 3792 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
 |- 
 |^| ( A  u.  B )  =  ( |^| A  i^i  |^| B )
 
Theoremintpr 3793 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| { A ,  B }  =  ( A  i^i  B )
 
Theoremintprg 3794 The intersection of a pair is the intersection of its members. Closed form of intpr 3793. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
 
Theoremintsng 3795 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  -> 
 |^| { A }  =  A )
 
Theoremintsn 3796 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
 |-  A  e.  _V   =>    |-  |^| { A }  =  A
 
Theoremuniintsn 3797* Two ways to express " A is a singleton." See also en1 6813, en1b 6814, card1 7485, and eusn 3607. (Contributed by NM, 2-Aug-2010.)
 |-  ( U. A  =  |^|
 A 
 <-> 
 E. x  A  =  { x } )
 
Theoremuniintab 3798 The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of  ph ( x ). (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  <->  U. { x  |  ph }  =  |^|
 { x  |  ph } )
 
Theoremintunsn 3799 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
 |-  B  e.  _V   =>    |-  |^| ( A  u.  { B } )  =  ( |^| A  i^i  B )
 
Theoremrint0 3800 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
    < 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 30901-30955
  Copyright terms: Public domain < Previous  Next >