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Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoteq123d 4001 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 4002 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x <. A ,  B >.
 
Theoremnfopd 4003 Deduction version of bound-variable hypothesis builder nfop 4002. This shows how the deduction version of a not-free theorem such as nfop 4002 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 4004 The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 |-  A  e.  _V   =>    |-  <. A ,  A >.  =  { { A } }
 
Theoremralunsn 4005* 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 4006* 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 4007 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 4008 Expansion of an ordered pair when the first member is a proper class. See also opprc 4007. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  A  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremopprc2 4009 Expansion of an ordered pair when the second member is a proper class. See also opprc 4007. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  B  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremoprcl 4010 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 4011 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
 |- 
 ~P { A }  =  { (/) ,  { A } }
 
TheorempwsnALT 4012 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 4013 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
 |- 
 ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
 
Theorempwtp 4014 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 4015 Compute the power set of the power set of the power set of the empty set. (See also pw0 3947 and pwpw0 3948.) (Contributed by NM, 2-May-2009.)
 |- 
 ~P { (/) ,  { (/)
 } }  =  ( { (/) ,  { (/) } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } } )
 
Theorempwv 4016 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 4017 Extend class notation to include the union of a class (read: 'union  A')
 class  U. A
 
Definitiondf-uni 4018* 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 21736). This is similar to the union of two classes df-un 3327. (Contributed by NM, 23-Aug-1993.)
 |- 
 U. A  =  { x  |  E. y
 ( x  e.  y  /\  y  e.  A ) }
 
Theoremdfuni2 4019* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
 |- 
 U. A  =  { x  |  E. y  e.  A  x  e.  y }
 
Theoremeluni 4020* Membership in class union. (Contributed by NM, 22-May-1994.)
 |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremeluni2 4021* 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 4022 Membership in class union. (Contributed by NM, 24-Mar-1995.)
 |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C )
 
Theoremnfuni 4023 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 4024 Deduction version of nfuni 4023. (Contributed by NM, 18-Feb-2013.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x U. A )
 
Theoremcsbunig 4025 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 4026 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 4027 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  U. A  =  U. B
 
Theoremunieqd 4028 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  U. A  =  U. B )
 
Theoremeluniab 4029* 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 4030* 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 4031 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 4032 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 4033 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 4034 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 4035* 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 4036 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 4037 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 6986 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 4038 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 4039 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 4040 Subclass relationship for subclass union. Inference form of uniss 4038. (Contributed by David Moews, 1-May-2017.)
 |-  A  C_  B   =>    |- 
 U. A  C_  U. B
 
Theoremunissd 4041 Subclass relationship for subclass union. Deduction form of uniss 4038. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  U. A  C_ 
 U. B )
 
Theoremuni0b 4042 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 4043* 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 4044 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4340 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
 |- 
 U. (/)  =  (/)
 
Theoremelssuni 4045 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 4046 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 4047* 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 4048* 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 4138 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 4049* 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 4050* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  x  C_  A )  ->  A  =  U. B )
 
Theoremunimax 4051* 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 4052 Extend class notation to include the intersection of a class (read: 'intersect  A').
 class  |^| A
 
Definitiondf-int 4053* 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 3329. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
 
Theoremdfint2 4054* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  { x  |  A. y  e.  A  x  e.  y }
 
Theoreminteq 4055 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
 |-  ( A  =  B  -> 
 |^| A  =  |^| B )
 
Theoreminteqi 4056 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  A  =  B   =>    |-  |^| A  =  |^| B
 
Theoreminteqd 4057 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  |^| A  =  |^| B )
 
Theoremelint 4058* 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 4059* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x  e.  B  A  e.  x )
 
Theoremelintg 4060* 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 4061 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 4062 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 4063* 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 4064* 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 4065* 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 4066 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| (/)  =  _V
 
Theoremintss1 4067 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 4068* 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 4069* 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 4070* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
 |-  A  C_  |^| { x  e.  B  |  A  C_  x }
 
Theoremssmin 4071* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
 |-  A  C_  |^| { x  |  ( A  C_  x  /\  ph ) }
 
Theoremintmin 4072* 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 4073 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  -> 
 |^| B  C_  |^| A )
 
Theoremintssuni 4074 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
 |-  ( A  =/=  (/)  ->  |^| A  C_ 
 U. A )
 
Theoremssintrab 4075* 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 4076 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4089). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( U. A  C_  |^|
 A 
 <->  ( A  =  (/)  \/ 
 U. A  =  |^| A ) )
 
Theoremintssuni2 4077 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  |^| A  C_  U. B )
 
Theoremintminss 4078* 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 4079* 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 4080* 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 4081* 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 4082* 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 6003 or abexssex 6004 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 4083 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  e.  A  -> 
 |^| A  =  (/) )
 
Theoremintun 4084 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 4085 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 4086 The intersection of a pair is the intersection of its members. Closed form of intpr 4085. 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 4087 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  -> 
 |^| { A }  =  A )
 
Theoremintsn 4088 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 4089* Two ways to express " A is a singleton." See also en1 7176, en1b 7177, card1 7857, and eusn 3882. (Contributed by NM, 2-Aug-2010.)
 |-  ( U. A  =  |^|
 A 
 <-> 
 E. x  A  =  { x } )
 
Theoremuniintab 4090 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 4091 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
 |-  B  e.  _V   =>    |-  |^| ( A  u.  { B } )  =  ( |^| A  i^i  B )
 
Theoremrint0 4092 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
 
Theoremelrint 4093* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y
 ) )
 
Theoremelrint2 4094* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  e.  A  ->  ( X  e.  ( A  i^i  |^| B )  <->  A. y  e.  B  X  e.  y )
 )
 
2.1.20  Indexed union and intersection
 
Syntaxciun 4095 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation  U. x  e.  A B, with the same union symbol as cuni 4017. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol  U_ instead of  U. and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
 class  U_ x  e.  A  B
 
Syntaxciin 4096 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation  |^| x  e.  A B, with the same intersection symbol as cint 4052. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol  |^|_ instead of  |^| and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
 class  |^|_
 x  e.  A  B
 
Definitiondf-iun 4097* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications,  A is independent of  x (although this is not required by the definition), and  B depends on  x i.e. can be read informally as  B ( x ). We call  x the index,  A the index set, and  B the indexed set. In most books,  x  e.  A is written as a subscript or underneath a union symbol  U.. We use a special union symbol  U_ to make it easier to distinguish from plain class union. In many theorems, you will see that  x and 
A are in the same distinct variable group (meaning  A cannot depend on  x) and that  B and  x do not share a distinct variable group (meaning that can be thought of as  B ( x ) i.e. can be substituted with a class expression containing 
x). An alternate definition tying indexed union to ordinary union is dfiun2 4127. Theorem uniiun 4146 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5996 and funiunfv 5997 are useful when  B is a function. (Contributed by NM, 27-Jun-1998.)
 |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
 
Definitiondf-iin 4098* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4097. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4128. Theorem intiin 4147 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
 |-  |^|_ x  e.  A  B  =  { y  |  A. x  e.  A  y  e.  B }
 
Theoremeliun 4099* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  U_ x  e.  B  C  <->  E. x  e.  B  A  e.  C )
 
Theoremeliin 4100* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  V  ->  ( A  e.  |^|_ x  e.  B  C  <->  A. x  e.  B  A  e.  C )
 )
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