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Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempreqr2g 3501 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3503. (Contributed by Jim Kingdon, 21-Sep-2018.)
 _V  _V  { C ,  }  { C ,  }
 
Theorempreqr1 3502 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.)
 _V   &     _V   =>     { ,  C }  { ,  C }
 
Theorempreqr2 3503 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.)
 _V   &     _V   =>     { C ,  }  { C ,  }
 
Theorempreq12b 3504 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
 _V   &     _V   &     C  _V   &     D  _V   =>     { ,  }  { C ,  D }  C  D  D  C
 
Theoremprel12 3505 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
 _V   &     _V   &     C  _V   &     D  _V   =>     { ,  }  { C ,  D }  { C ,  D }  { C ,  D }
 
Theoremopthpr 3506 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
 _V   &     _V   &     C  _V   &     D  _V   =>     =/=  D  { ,  }  { C ,  D }  C  D
 
Theorempreq12bg 3507 Closed form of preq12b 3504. (Contributed by Scott Fenton, 28-Mar-2014.)
 V  W  C  X  D  Y  { ,  }  { C ,  D }  C  D  D  C
 
Theoremprneimg 3508 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.)
 U  V  C  X  D  Y  =/=  C  =/=  D  =/=  C  =/=  D  { ,  }  =/=  { C ,  D }
 
Theorempreqsn 3509 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
 _V   &     _V   &     C  _V   =>     { ,  }  { C }  C
 
Theoremdfopg 3510 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
 V  W  <. ,  >.  { { } ,  { ,  } }
 
Theoremdfop 3511 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.)
 _V   &     _V   =>     <. ,  >.  { { } ,  { ,  } }
 
Theoremopeq1 3512 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 <. ,  C >.  <. ,  C >.
 
Theoremopeq2 3513 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 <. C ,  >.  <. C ,  >.
 
Theoremopeq12 3514 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
 C  D  <. ,  >.  <. C ,  D >.
 
Theoremopeq1i 3515 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
   =>    
 <. ,  C >. 
 <. ,  C >.
 
Theoremopeq2i 3516 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
   =>    
 <. C ,  >. 
 <. C ,  >.
 
Theoremopeq12i 3517 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
   &     C  D   =>     <. ,  C >.  <. ,  D >.
 
Theoremopeq1d 3518 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
   =>     <. ,  C >.  <. ,  C >.
 
Theoremopeq2d 3519 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
   =>     <. C ,  >.  <. C ,  >.
 
Theoremopeq12d 3520 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
   &     C  D   =>     <. ,  C >.  <. ,  D >.
 
Theoremoteq1 3521 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 <. ,  C ,  D >.  <. ,  C ,  D >.
 
Theoremoteq2 3522 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 <. C ,  ,  D >.  <. C ,  ,  D >.
 
Theoremoteq3 3523 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
 <. C ,  D ,  >.  <. C ,  D ,  >.
 
Theoremoteq1d 3524 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
   =>     <. ,  C ,  D >.  <. ,  C ,  D >.
 
Theoremoteq2d 3525 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
   =>     <. C ,  ,  D >.  <. C ,  ,  D >.
 
Theoremoteq3d 3526 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
   =>     <. C ,  D ,  >.  <. C ,  D ,  >.
 
Theoremoteq123d 3527 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
   &     C  D   &     E  F   =>     <. ,  C ,  E >.  <. ,  D ,  F >.
 
Theoremnfop 3528 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
 F/_   &     F/_   =>     F/_ <. ,  >.
 
Theoremnfopd 3529 Deduction version of bound-variable hypothesis builder nfop 3528. This shows how the deduction version of a not-free theorem such as nfop 3528 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
 F/_   &     F/_   =>     F/_ <. ,  >.
 
Theoremopid 3530 The ordered pair  <. ,  >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
 _V   =>    
 <. ,  >. 
 { { } }
 
Theoremralunsn 3531* 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.)
   =>     C  u.  { }
 
Theorem2ralunsn 3532* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
   &       &       =>     C  u.  { }  u.  { }
 
Theoremopprc 3533 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
 _V  _V  <. ,  >.  (/)
 
Theoremopprc1 3534 Expansion of an ordered pair when the first member is a proper class. See also opprc 3533. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V  <. ,  >.  (/)
 
Theoremopprc2 3535 Expansion of an ordered pair when the second member is a proper class. See also opprc 3533. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V  <. ,  >.  (/)
 
Theoremoprcl 3536 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
 C  <. ,  >.  _V  _V
 
Theorempwsnss 3537 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)

 { (/) ,  { } }  C_  ~P { }
 
Theorempwpw0ss 3538 Compute the power set of the power set of the empty set. (See pw0 3474 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 3539 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
 { (/) ,  { } }  u.  { { } ,  { ,  } }  C_  ~P { ,  }
 
Theorempwtpss 3540 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
 { (/) ,  { } }  u.  { { } ,  { ,  } }  u.  { { C } ,  { ,  C } }  u.  { { ,  C } ,  { ,  ,  C } }  C_  ~P { ,  ,  C }
 
Theorempwpwpw0ss 3541 Compute the power set of the power set of the power set of the empty set. (See also pw0 3474 and pwpw0ss 3538.) (Contributed by Jim Kingdon, 13-Aug-2018.)
 { (/) ,  { (/)
 } }  u.  { { { (/) } } ,  { (/) ,  { (/) } } }  C_  ~P { (/)
 ,  { (/) } }
 
Theorempwv 3542 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 3543 Extend class notation to include the union of a class (read: 'union ')
 U.
 
Definitiondf-uni 3544* 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 2890. (Contributed by NM, 23-Aug-1993.)

 U.  {  |  }
 
Theoremdfuni2 3545* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)

 U.  {  |  }
 
Theoremeluni 3546* Membership in class union. (Contributed by NM, 22-May-1994.)
 U.
 
Theoremeluni2 3547* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
 U.
 
Theoremelunii 3548 Membership in class union. (Contributed by NM, 24-Mar-1995.)
 C  U. C
 
Theoremnfuni 3549 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 F/_   =>     F/_ U.
 
Theoremnfunid 3550 Deduction version of nfuni 3549. (Contributed by NM, 18-Feb-2013.)
 F/_   =>     F/_ U.
 
Theoremcsbunig 3551 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 V  [_  ]_
 U.  U. [_  ]_
 
Theoremunieq 3552 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.)
 U.  U.
 
Theoremunieqi 3553 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
   =>    
 U.  U.
 
Theoremunieqd 3554 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
   =>     U.  U.
 
Theoremeluniab 3555* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
 U. {  |  }
 
Theoremelunirab 3556* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
 U. {  | 
 }
 
Theoremunipr 3557 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
 _V   &     _V   =>     U. { ,  }  u.
 
Theoremuniprg 3558 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
 V  W  U. { ,  }  u.
 
Theoremunisn 3559 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
 _V   =>    
 U. { }
 
Theoremunisng 3560 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
 V  U. { }
 
Theoremdfnfc2 3561* An alternative statement of the effective freeness of a class , when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
 V  F/_  F/
 
Theoremuniun 3562 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

 U.  u.  U.  u.  U.
 
Theoremuniin 3563 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.  i^i  C_  U.  i^i  U.
 
Theoremuniss 3564 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.)
 C_  U.  C_  U.
 
Theoremssuni 3565 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 C_  C  C_  U. C
 
Theoremunissi 3566 Subclass relationship for subclass union. Inference form of uniss 3564. (Contributed by David Moews, 1-May-2017.)
 C_    =>    
 U.  C_  U.
 
Theoremunissd 3567 Subclass relationship for subclass union. Deduction form of uniss 3564. (Contributed by David Moews, 1-May-2017.)
 C_    =>     U.  C_  U.
 
Theoremuni0b 3568 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.  (/)  C_  { (/) }
 
Theoremuni0c 3569* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
 U.  (/)  (/)
 
Theoremuni0 3570 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 3571 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.)
 C_  U.
 
Theoremunissel 3572 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
 U.  C_  U.
 
Theoremunissb 3573* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
 U.  C_  C_
 
Theoremuniss2 3574* 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.)
 C_  U.  C_  U.
 
Theoremunidif 3575* If the difference  \ contains the largest members of , then the union of the difference is the union of . (Contributed by NM, 22-Mar-2004.)
 \  C_  U.  \ 
 U.
 
Theoremssunieq 3576* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
 C_  U.
 
Theoremunimax 3577* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
 U. {  |  C_  }
 
2.1.19  The intersection of a class
 
Syntaxcint 3578 Extend class notation to include the intersection of a class (read: 'intersect ').
 |^|
 
Definitiondf-int 3579* 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 2892. (Contributed by NM, 18-Aug-1993.)

 |^|  {  |  }
 
Theoremdfint2 3580* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

 |^|  {  |  }
 
Theoreminteq 3581 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
 |^|  |^|
 
Theoreminteqi 3582 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
   =>    
 |^|  |^|
 
Theoreminteqd 3583 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
   =>     |^|  |^|
 
Theoremelint 3584* Membership in class intersection. (Contributed by NM, 21-May-1994.)
 _V   =>     |^|
 
Theoremelint2 3585* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
 _V   =>     |^|
 
Theoremelintg 3586* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
 V  |^|
 
Theoremelinti 3587 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |^|  C  C
 
Theoremnfint 3588 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 F/_   =>     F/_ |^|
 
Theoremelintab 3589* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
 _V   =>     |^| {  |  }
 
Theoremelintrab 3590* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
 _V   =>     |^| {  | 
 }
 
Theoremelintrabg 3591* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
 V  |^| {  | 
 }
 
Theoremint0 3592 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

 |^| (/)  _V
 
Theoremintss1 3593 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.)
 |^|  C_
 
Theoremssint 3594* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
 C_  |^|  C_
 
Theoremssintab 3595* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 C_  |^| {  |  }  C_
 
Theoremssintub 3596* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
 C_  |^| {  |  C_  }
 
Theoremssmin 3597* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
 C_  |^| {  |  C_  }
 
Theoremintmin 3598* 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.)
 |^| {  |  C_  }
 
Theoremintss 3599 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
 C_  |^|  C_  |^|
 
Theoremintssunim 3600* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
 |^|  C_ 
 U.
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