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Definition df-en 6642
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 6648. (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 6639 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1331 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1331 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1331 . . . . 5  class  f
83, 5, 7wf1o 5129 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1469 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 3995 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1332 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6645  bren  6648  enssdom  6663
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