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Definition df-en 7510
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 7518. (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 7506 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1397 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1397 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1397 . . . . 5  class  f
83, 5, 7wf1o 5569 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1617 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4496 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1398 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff setvar class
This definition is referenced by:  relen  7514  bren  7518  enssdom  7533
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