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Definition df-en 6754
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 6760. (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 6751 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1362 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1362 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1362 . . . . 5  class  f
83, 5, 7wf1o 5227 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1502 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4075 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1363 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6757  bren  6760  enssdom  6775
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