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Definition df-en 6866
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 6873. (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 6862 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1624 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1624 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1624 . . . . 5  class  f
83, 5, 7wf1o 5256 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1530 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4078 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1625 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6870  bren  6873  enssdom  6888
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