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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremixpm 6501* If an infinite Cartesian product of a family  B ( x ) is inhabited, every  B ( x ) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
 |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  E. z  z  e.  B )
 
Theoremixp0 6502 The infinite Cartesian product of a family  B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
 |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
 
Theoremixpssmap 6503* An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
 |-  B  e.  _V   =>    |-  X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A )
 
Theoremresixp 6504* Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  ( ( B  C_  A  /\  F  e.  X_ x  e.  A  C )  ->  ( F  |`  B )  e.  X_ x  e.  B  C )
 
Theoremmptelixpg 6505* Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
 |-  ( I  e.  V  ->  ( ( x  e.  I  |->  J )  e.  X_ x  e.  I  K 
 <-> 
 A. x  e.  I  J  e.  K )
 )
 
Theoremelixpsn 6506* Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e.  V  ->  ( F  e.  X_ x  e.  { A } B  <->  E. y  e.  B  F  =  { <. A ,  y >. } ) )
 
Theoremixpsnf1o 6507* A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )   =>    |-  ( I  e.  V  ->  F : A
 -1-1-onto-> X_ y  e.  { I } A )
 
Theoremmapsnf1o 6508* A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )   =>    |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> ( A  ^m  { I } ) )
 
2.6.27  Equinumerosity
 
Syntaxcen 6509 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
 class  ~~
 
Syntaxcdom 6510 Extend class definition to include the dominance relation (curly less-than-or-equal)
 class  ~<_
 
Syntaxcfn 6511 Extend class definition to include the class of all finite sets.
 class  Fin
 
Definitiondf-en 6512* Define the equinumerosity relation. Definition of [Enderton] p. 129. We define  ~~ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 6518. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
 
Definitiondf-dom 6513* Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6521 and domen 6522. (Contributed by NM, 28-Mar-1998.)
 |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
 
Definitiondf-fin 6514* Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our " a  e.  Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 12137. (Contributed by NM, 22-Aug-2008.)
 |- 
 Fin  =  { x  |  E. y  e.  om  x  ~~  y }
 
Theoremrelen 6515 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~~
 
Theoremreldom 6516 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~<_
 
Theoremencv 6517 If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
 |-  ( A  ~~  B  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theorembren 6518* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B )
 
Theorembrdomg 6519* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
 
Theorembrdomi 6520* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~<_  B  ->  E. f  f : A -1-1-> B )
 
Theorembrdom 6521* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
 
Theoremdomen 6522* Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. x ( A 
 ~~  x  /\  x  C_  B ) )
 
Theoremdomeng 6523* Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x ( A 
 ~~  x  /\  x  C_  B ) ) )
 
Theoremctex 6524 A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A  ~<_  om  ->  A  e.  _V )
 
Theoremf1oen3g 6525 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6528 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
 
Theoremf1oen2g 6526 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6528 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
 
Theoremf1dom2g 6527 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6529 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )
 
Theoremf1oeng 6528 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
 |-  ( ( A  e.  C  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
 
Theoremf1domg 6529 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
 |-  ( B  e.  C  ->  ( F : A -1-1-> B 
 ->  A  ~<_  B ) )
 
Theoremf1oen 6530 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
 |-  A  e.  _V   =>    |-  ( F : A
 -1-1-onto-> B  ->  A  ~~  B )
 
Theoremf1dom 6531 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( F : A -1-1-> B  ->  A  ~<_  B )
 
Theoremisfi 6532* Express " A is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose " Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
 |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x )
 
Theoremenssdom 6533 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~~  C_  ~<_
 
Theoremendom 6534 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
 |-  ( A  ~~  B  ->  A  ~<_  B )
 
Theoremenrefg 6535 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  ->  A  ~~  A )
 
Theoremenref 6536 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  e.  _V   =>    |-  A  ~~  A
 
Theoremeqeng 6537 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
 |-  ( A  e.  V  ->  ( A  =  B  ->  A  ~~  B ) )
 
Theoremdomrefg 6538 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
 |-  ( A  e.  V  ->  A  ~<_  A )
 
Theoremen2d 6539* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V ) )   &    |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  A  ~~  B )
 
Theoremen3d 6540* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <->  y  =  C ) ) )   =>    |-  ( ph  ->  A 
 ~~  B )
 
Theoremen2i 6541* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  e.  A  ->  C  e.  _V )   &    |-  ( y  e.  B  ->  D  e.  _V )   &    |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) )   =>    |-  A  ~~  B
 
Theoremen3i 6542* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( y  e.  B  ->  D  e.  A )   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  A  ~~  B
 
Theoremdom2lem 6543* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
 
Theoremdom2d 6544* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   =>    |-  ( ph  ->  ( B  e.  R  ->  A  ~<_  B ) )
 
Theoremdom3d 6545* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  A  ~<_  B )
 
Theoremdom2 6546* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
 |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y
 ) )   =>    |-  ( B  e.  V  ->  A  ~<_  B )
 
Theoremdom3 6547* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
 |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y
 ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
 
Theoremidssen 6548 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 _I  C_  ~~
 
Theoremssdomg 6549 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( B  e.  V  ->  ( A  C_  B  ->  A  ~<_  B ) )
 
Theoremener 6550 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 ~~  Er  _V
 
Theoremensymb 6551 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~~  B  <->  B 
 ~~  A )
 
Theoremensym 6552 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~~  B  ->  B  ~~  A )
 
Theoremensymi 6553 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   =>    |-  B  ~~  A
 
Theoremensymd 6554 Symmetry of equinumerosity. Deduction form of ensym 6552. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  ~~  B )   =>    |-  ( ph  ->  B  ~~  A )
 
Theorementr 6555 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
 
Theoremdomtr 6556 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  ~<_  B  /\  B 
 ~<_  C )  ->  A  ~<_  C )
 
Theorementri 6557 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  A  ~~  C
 
Theorementr2i 6558 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  C  ~~  A
 
Theorementr3i 6559 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  A  ~~  C   =>    |-  B  ~~  C
 
Theorementr4i 6560 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  C  ~~  B   =>    |-  A  ~~  C
 
Theoremendomtr 6561 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~<_  C ) 
 ->  A  ~<_  C )
 
Theoremdomentr 6562 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A 
 ~<_  C )
 
Theoremf1imaeng 6563 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C )  ~~  C )
 
Theoremf1imaen2g 6564 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6565 does not need ax-setind 4366.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( ( F : A -1-1-> B  /\  B  e.  V )  /\  ( C  C_  A  /\  C  e.  V ) )  ->  ( F " C )  ~~  C )
 
Theoremf1imaen 6565 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
 |-  C  e.  _V   =>    |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C ) 
 ~~  C )
 
Theoremen0 6566 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
 |-  ( A  ~~  (/)  <->  A  =  (/) )
 
Theoremensn1 6567 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
 |-  A  e.  _V   =>    |-  { A }  ~~  1o
 
Theoremensn1g 6568 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  { A }  ~~  1o )
 
Theoremenpr1g 6569  { A ,  A } has only one element. (Contributed by FL, 15-Feb-2010.)
 |-  ( A  e.  V  ->  { A ,  A }  ~~  1o )
 
Theoremen1 6570* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
 |-  ( A  ~~  1o  <->  E. x  A  =  { x } )
 
Theoremen1bg 6571 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
 |-  ( A  e.  V  ->  ( A  ~~  1o  <->  A  =  { U. A }
 ) )
 
Theoremreuen1 6572* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x  e.  A  ph  <->  { x  e.  A  |  ph }  ~~  1o )
 
Theoremeuen1 6573 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )
 
Theoremeuen1b 6574* Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  1o  <->  E! x  x  e.  A )
 
Theoremen1uniel 6575 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( S  ~~  1o  ->  U. S  e.  S )
 
Theorem2dom 6576* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
 |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
 
Theoremfundmen 6577 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  F  e.  _V   =>    |-  ( Fun  F  ->  dom  F  ~~  F )
 
Theoremfundmeng 6578 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
 |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F  ~~  F )
 
Theoremcnven 6579 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( ( Rel  A  /\  A  e.  V ) 
 ->  A  ~~  `' A )
 
Theoremcnvct 6580 If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A  ~<_  om  ->  `' A  ~<_  om )
 
Theoremfndmeng 6581 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
 
Theoremmapsnen 6582 Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  { B } )  ~~  A
 
Theoremmap1 6583 Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)
 |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
 
Theoremen2sn 6584 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A }  ~~  { B } )
 
Theoremsnfig 6585 A singleton is finite. For the proper class case, see snprc 3511. (Contributed by Jim Kingdon, 13-Apr-2020.)
 |-  ( A  e.  V  ->  { A }  e.  Fin )
 
Theoremfiprc 6586 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
 |- 
 Fin  e/  _V
 
Theoremunen 6587 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A 
 ~~  B  /\  C  ~~  D )  /\  (
 ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D ) )
 
Theoremssct 6588 Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theorem1domsn 6589 A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
 |- 
 { A }  ~<_  1o
 
Theoremenm 6590* A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  ( ( A  ~~  B  /\  E. x  x  e.  A )  ->  E. y  y  e.  B )
 
Theoremxpsnen 6591 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  { B } )  ~~  A
 
Theoremxpsneng 6592 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A )
 
Theoremxp1en 6593 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
 
Theoremendisj 6594* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E. x E. y
 ( ( x  ~~  A  /\  y  ~~  B )  /\  ( x  i^i  y )  =  (/) )
 
Theoremxpcomf1o 6595* The canonical bijection from  ( A  X.  B
) to  ( B  X.  A ). (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   =>    |-  F : ( A  X.  B ) -1-1-onto-> ( B  X.  A )
 
Theoremxpcomco 6596* Composition with the bijection of xpcomf1o 6595 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   &    |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )   =>    |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
 
Theoremxpcomen 6597 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  B )  ~~  ( B  X.  A )
 
Theoremxpcomeng 6598 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
 
Theoremxpsnen2g 6599 A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B )
 
Theoremxpassen 6600 Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  X.  C )  ~~  ( A  X.  ( B  X.  C ) )
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