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Definition df-en 7047
Description: 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 7054. (Contributed by NM, 28-Mar-1998.)
Assertion
Ref Expression
df-en  |-  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Distinct variable group:    x, y, f

Detailed syntax breakdown of Definition df-en
StepHypRef Expression
1 cen 7043 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1648 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1648 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1648 . . . . 5  class  f
83, 5, 7wf1o 5394 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1547 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4207 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1649 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  7051  bren  7054  enssdom  7069
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