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Definition df-en 7112
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 7119. (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 7108 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1652 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1652 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1652 . . . . 5  class  f
83, 5, 7wf1o 5455 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1551 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4267 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1653 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  7116  bren  7119  enssdom  7134
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