HomeHome Intuitionistic Logic Explorer
Theorem List (p. 37 of 114)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprneimg 3601 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 }
 ) )
 
Theorempreqsn 3602 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 ) )
 
Theoremdfopg 3603 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 3604 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 3605 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 3606 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 3607 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  <. A ,  B >.  =  <. C ,  D >. )
 
Theoremopeq1i 3608 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. A ,  C >.  =  <. B ,  C >.
 
Theoremopeq2i 3609 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  A  =  B   =>    |-  <. C ,  A >.  =  <. C ,  B >.
 
Theoremopeq12i 3610 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 3611 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C >.  =  <. B ,  C >. )
 
Theoremopeq2d 3612 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A >.  =  <. C ,  B >. )
 
Theoremopeq12d 3613 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 3614 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. A ,  C ,  D >.  =  <. B ,  C ,  D >. )
 
Theoremoteq2 3615 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  A ,  D >.  =  <. C ,  B ,  D >. )
 
Theoremoteq3 3616 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 |-  ( A  =  B  -> 
 <. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
 
Theoremoteq1d 3617 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. A ,  C ,  D >.  = 
 <. B ,  C ,  D >. )
 
Theoremoteq2d 3618 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  A ,  D >.  = 
 <. C ,  B ,  D >. )
 
Theoremoteq3d 3619 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <. C ,  D ,  A >.  = 
 <. C ,  D ,  B >. )
 
Theoremoteq123d 3620 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 3621 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x <. A ,  B >.
 
Theoremnfopd 3622 Deduction version of bound-variable hypothesis builder nfop 3621. This shows how the deduction version of a not-free theorem such as nfop 3621 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 3623 The ordered pair  <. A ,  A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 |-  A  e.  _V   =>    |-  <. A ,  A >.  =  { { A } }
 
Theoremralunsn 3624* 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 3625* 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 3626 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 3627 Expansion of an ordered pair when the first member is a proper class. See also opprc 3626. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  A  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremopprc2 3628 Expansion of an ordered pair when the second member is a proper class. See also opprc 3626. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( -.  B  e.  _V 
 ->  <. A ,  B >.  =  (/) )
 
Theoremoprcl 3629 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 ) )
 
Theorempwsnss 3630 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
 |- 
 { (/) ,  { A } }  C_  ~P { A }
 
Theorempwpw0ss 3631 Compute the power set of the power set of the empty set. (See pw0 3567 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)
 |- 
 { (/) ,  { (/) } }  C_ 
 ~P { (/) }
 
Theorempwprss 3632 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
 |-  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )  C_  ~P { A ,  B }
 
Theorempwtpss 3633 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
 |-  ( ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )  u.  ( { { C } ,  { A ,  C } }  u.  { { B ,  C } ,  { A ,  B ,  C } } ) ) 
 C_  ~P { A ,  B ,  C }
 
Theorempwpwpw0ss 3634 Compute the power set of the power set of the power set of the empty set. (See also pw0 3567 and pwpw0ss 3631.) (Contributed by Jim Kingdon, 13-Aug-2018.)
 |-  ( { (/) ,  { (/)
 } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } } )  C_  ~P { (/)
 ,  { (/) } }
 
Theorempwv 3635 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 3636 Extend class notation to include the union of a class (read: 'union  A')
 class  U. A
 
Definitiondf-uni 3637* 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, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to the union of two classes df-un 2992. (Contributed by NM, 23-Aug-1993.)
 |- 
 U. A  =  { x  |  E. y
 ( x  e.  y  /\  y  e.  A ) }
 
Theoremdfuni2 3638* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
 |- 
 U. A  =  { x  |  E. y  e.  A  x  e.  y }
 
Theoremeluni 3639* Membership in class union. (Contributed by NM, 22-May-1994.)
 |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremeluni2 3640* 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 3641 Membership in class union. (Contributed by NM, 24-Mar-1995.)
 |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C )
 
Theoremnfuni 3642 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 3643 Deduction version of nfuni 3642. (Contributed by NM, 18-Feb-2013.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x U. A )
 
Theoremcsbunig 3644 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 3645 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 3646 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  U. A  =  U. B
 
Theoremunieqd 3647 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  U. A  =  U. B )
 
Theoremeluniab 3648* 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 3649* 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 3650 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 3651 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 3652 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 3653 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 3654* An alternate 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 3655 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 3656 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (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 3657 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 3658 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 3659 Subclass relationship for subclass union. Inference form of uniss 3657. (Contributed by David Moews, 1-May-2017.)
 |-  A  C_  B   =>    |- 
 U. A  C_  U. B
 
Theoremunissd 3660 Subclass relationship for subclass union. Deduction form of uniss 3657. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  U. A  C_ 
 U. B )
 
Theoremuni0b 3661 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 3662* 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 3663 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
 |- 
 U. (/)  =  (/)
 
Theoremelssuni 3664 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 3665 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 3666* 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 3667* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)
 |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  U. A  C_  U. B )
 
Theoremunidif 3668* 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 3669* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  x  C_  A )  ->  A  =  U. B )
 
Theoremunimax 3670* 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 3671 Extend class notation to include the intersection of a class (read: 'intersect  A').
 class  |^| A
 
Definitiondf-int 3672* 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 2994. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
 
Theoremdfint2 3673* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  { x  |  A. y  e.  A  x  e.  y }
 
Theoreminteq 3674 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
 |-  ( A  =  B  -> 
 |^| A  =  |^| B )
 
Theoreminteqi 3675 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  A  =  B   =>    |-  |^| A  =  |^| B
 
Theoreminteqd 3676 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  |^| A  =  |^| B )
 
Theoremelint 3677* 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 3678* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
 |-  A  e.  _V   =>    |-  ( A  e.  |^|
 B 
 <-> 
 A. x  e.  B  A  e.  x )
 
Theoremelintg 3679* 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 3680 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 3681 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 3682* 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 3683* 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 3684* 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 3685 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
 |- 
 |^| (/)  =  _V
 
Theoremintss1 3686 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 3687* 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 3688* 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 3689* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
 |-  A  C_  |^| { x  e.  B  |  A  C_  x }
 
Theoremssmin 3690* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
 |-  A  C_  |^| { x  |  ( A  C_  x  /\  ph ) }
 
Theoremintmin 3691* 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 3692 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  -> 
 |^| B  C_  |^| A )
 
Theoremintssunim 3693* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
 |-  ( E. x  x  e.  A  ->  |^| A  C_ 
 U. A )
 
Theoremssintrab 3694* 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 ) )
 
Theoremintssuni2m 3695* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
 |-  ( ( A  C_  B  /\  E. x  x  e.  A )  ->  |^| A  C_  U. B )
 
Theoremintminss 3696* 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 3697* 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 3698* 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 3699* 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 3700* 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 ). (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 ) }
    < 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 11301-11363
  Copyright terms: Public domain < Previous  Next >