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Definition df-en 7569
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 7577. (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 7565 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1436 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1436 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1436 . . . . 5  class  f
83, 5, 7wf1o 5591 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1659 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4474 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1437 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff setvar class
This definition is referenced by:  relen  7573  bren  7577  enssdom  7592
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