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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelpmg 6401 The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  ( A  ^pm  B )  <-> 
 ( Fun  C  /\  C  C_  ( B  X.  A ) ) ) )
 
Theoremelpm2g 6402 The predicate "is a partial function." (Contributed by NM, 31-Dec-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F  e.  ( A  ^pm  B )  <-> 
 ( F : dom  F --> A  /\  dom  F  C_  B ) ) )
 
Theoremelpm2r 6403 Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( F : C --> A  /\  C  C_  B ) ) 
 ->  F  e.  ( A 
 ^pm  B ) )
 
Theoremelpmi 6404 A partial function is a function. (Contributed by Mario Carneiro, 15-Sep-2015.)
 |-  ( F  e.  ( A  ^pm  B )  ->  ( F : dom  F --> A  /\  dom  F  C_  B ) )
 
Theorempmfun 6405 A partial function is a function. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( F  e.  ( A  ^pm  B )  ->  Fun  F )
 
Theoremelmapex 6406 Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  ( A  e.  ( B  ^m  C )  ->  ( B  e.  _V  /\  C  e.  _V )
 )
 
Theoremelmapi 6407 A mapping is a function, forward direction only with superfluous antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
 
Theoremelmapfn 6408 A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
 |-  ( A  e.  ( B  ^m  C )  ->  A  Fn  C )
 
Theoremelmapfun 6409 A mapping is always a function. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( A  e.  ( B  ^m  C )  ->  Fun  A )
 
Theoremelmapssres 6410 A restricted mapping is a mapping. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( ( A  e.  ( B  ^m  C ) 
 /\  D  C_  C )  ->  ( A  |`  D )  e.  ( B  ^m  D ) )
 
Theoremfpmg 6411 A total function is a partial function. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A --> B ) 
 ->  F  e.  ( B 
 ^pm  A ) )
 
Theorempmss12g 6412 Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ( ( A 
 C_  C  /\  B  C_  D )  /\  ( C  e.  V  /\  D  e.  W )
 )  ->  ( A  ^pm 
 B )  C_  ( C  ^pm  D ) )
 
Theorempmresg 6413 Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
 |-  ( ( B  e.  V  /\  F  e.  ( A  ^pm  C ) ) 
 ->  ( F  |`  B )  e.  ( A  ^pm  B ) )
 
Theoremelmap 6414 Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F  e.  ( A  ^m  B )  <->  F : B --> A )
 
Theoremmapval2 6415* Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  B )  =  ( ~P ( B  X.  A )  i^i  { f  |  f  Fn  B }
 )
 
Theoremelpm 6416 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F  e.  ( A  ^pm  B )  <->  ( Fun  F  /\  F  C_  ( B  X.  A ) ) )
 
Theoremelpm2 6417 The predicate "is a partial function." (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F  e.  ( A  ^pm  B )  <->  ( F : dom  F --> A  /\  dom  F 
 C_  B ) )
 
Theoremfpm 6418 A total function is a partial function. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F : A --> B  ->  F  e.  ( B  ^pm  A ) )
 
Theoremmapsspm 6419 Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  ( A  ^m  B )  C_  ( A  ^pm  B )
 
Theorempmsspw 6420 Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( A  ^pm  B )  C_  ~P ( B  X.  A )
 
Theoremmapsspw 6421 Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ^m  B )  C_  ~P ( B  X.  A )
 
Theoremfvmptmap 6422* Special case of fvmpt 5365 for operator theorems. (Contributed by NM, 27-Nov-2007.)
 |-  C  e.  _V   &    |-  D  e.  _V   &    |-  R  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  ( R  ^m  D )  |->  B )   =>    |-  ( A : D --> R  ->  ( F `  A )  =  C )
 
Theoremmap0e 6423 Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
 
Theoremmap0b 6424 Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  =/=  (/)  ->  ( (/)  ^m  A )  =  (/) )
 
Theoremmap0g 6425 Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A 
 ^m  B )  =  (/) 
 <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
 
Theoremmap0 6426 Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) )
 
Theoremmapsn 6427* The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  { B } )  =  {
 f  |  E. y  e.  A  f  =  { <. B ,  y >. } }
 
Theoremmapss 6428 Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( B  e.  V  /\  A  C_  B )  ->  ( A  ^m  C )  C_  ( B 
 ^m  C ) )
 
Theoremfdiagfn 6429* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  B  |->  ( I  X.  { x }
 ) )   =>    |-  ( ( B  e.  V  /\  I  e.  W )  ->  F : B --> ( B  ^m  I ) )
 
Theoremfvdiagfn 6430* Functionality of the diagonal map. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  F  =  ( x  e.  B  |->  ( I  X.  { x }
 ) )   =>    |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( F `
  X )  =  ( I  X.  { X } ) )
 
Theoremmapsnconst 6431 Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   =>    |-  ( F  e.  ( B  ^m  S )  ->  F  =  ( S  X.  { ( F `  X ) } )
 )
 
Theoremmapsncnv 6432* Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   &    |-  F  =  ( x  e.  ( B 
 ^m  S )  |->  ( x `  X ) )   =>    |-  `' F  =  (
 y  e.  B  |->  ( S  X.  { y } ) )
 
Theoremmapsnf1o2 6433* Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   &    |-  F  =  ( x  e.  ( B 
 ^m  S )  |->  ( x `  X ) )   =>    |-  F : ( B 
 ^m  S ) -1-1-onto-> B
 
Theoremmapsnf1o3 6434* Explicit bijection in the reverse of mapsnf1o2 6433. (Contributed by Stefan O'Rear, 24-Mar-2015.)
 |-  S  =  { X }   &    |-  B  e.  _V   &    |-  X  e.  _V   &    |-  F  =  ( y  e.  B  |->  ( S  X.  { y } ) )   =>    |-  F : B -1-1-onto-> ( B  ^m  S )
 
2.6.26  Equinumerosity
 
Syntaxcen 6435 Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
 class  ~~
 
Syntaxcdom 6436 Extend class definition to include the dominance relation (curly less-than-or-equal)
 class  ~<_
 
Syntaxcfn 6437 Extend class definition to include the class of all finite sets.
 class  Fin
 
Definitiondf-en 6438* 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 6444. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
 
Definitiondf-dom 6439* Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6447 and domen 6448. (Contributed by NM, 28-Mar-1998.)
 |-  ~<_  =  { <. x ,  y >.  |  E. f  f : x -1-1-> y }
 
Definitiondf-fin 6440* 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 11528. (Contributed by NM, 22-Aug-2008.)
 |- 
 Fin  =  { x  |  E. y  e.  om  x  ~~  y }
 
Theoremrelen 6441 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~~
 
Theoremreldom 6442 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
 |- 
 Rel  ~<_
 
Theoremencv 6443 If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
 |-  ( A  ~~  B  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theorembren 6444* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( A  ~~  B  <->  E. f  f : A -1-1-onto-> B )
 
Theorembrdomg 6445* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
 
Theorembrdomi 6446* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~<_  B  ->  E. f  f : A -1-1-> B )
 
Theorembrdom 6447* Dominance relation. (Contributed by NM, 15-Jun-1998.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
 
Theoremdomen 6448* 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 6449* 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 6450 A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A  ~<_  om  ->  A  e.  _V )
 
Theoremf1oen3g 6451 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6454 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 6452 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6454 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 6453 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6455 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 6454 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 6455 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 6456 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 6457 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 6458* 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 6459 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~~  C_  ~<_
 
Theoremendom 6460 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
 |-  ( A  ~~  B  ->  A  ~<_  B )
 
Theoremenrefg 6461 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 6462 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  e.  _V   =>    |-  A  ~~  A
 
Theoremeqeng 6463 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
 |-  ( A  e.  V  ->  ( A  =  B  ->  A  ~~  B ) )
 
Theoremdomrefg 6464 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
 |-  ( A  e.  V  ->  A  ~<_  A )
 
Theoremen2d 6465* 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 6466* 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 6467* 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 6468* 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 6469* 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 6470* 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 6471* 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 6472* 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 6473* 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 6474 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 _I  C_  ~~
 
Theoremssdomg 6475 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 6476 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 ~~  Er  _V
 
Theoremensymb 6477 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~~  B  <->  B 
 ~~  A )
 
Theoremensym 6478 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 6479 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   =>    |-  B  ~~  A
 
Theoremensymd 6480 Symmetry of equinumerosity. Deduction form of ensym 6478. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  ~~  B )   =>    |-  ( ph  ->  B  ~~  A )
 
Theorementr 6481 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
 
Theoremdomtr 6482 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 6483 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  A  ~~  C
 
Theorementr2i 6484 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  C  ~~  A
 
Theorementr3i 6485 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  A  ~~  C   =>    |-  B  ~~  C
 
Theorementr4i 6486 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  C  ~~  B   =>    |-  A  ~~  C
 
Theoremendomtr 6487 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~<_  C ) 
 ->  A  ~<_  C )
 
Theoremdomentr 6488 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A 
 ~<_  C )
 
Theoremf1imaeng 6489 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 6490 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6491 does not need ax-setind 4343.) (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 6491 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 6492 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
 |-  ( A  ~~  (/)  <->  A  =  (/) )
 
Theoremensn1 6493 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
 |-  A  e.  _V   =>    |-  { A }  ~~  1o
 
Theoremensn1g 6494 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  { A }  ~~  1o )
 
Theoremenpr1g 6495  { A ,  A } has only one element. (Contributed by FL, 15-Feb-2010.)
 |-  ( A  e.  V  ->  { A ,  A }  ~~  1o )
 
Theoremen1 6496* 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 6497 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 6498* 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 6499 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )
 
Theoremeuen1b 6500* Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  1o  <->  E! x  x  e.  A )
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