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Definition df-en 6603
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 6609. (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 6600 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1315 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1315 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1315 . . . . 5  class  f
83, 5, 7wf1o 5092 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1453 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 3958 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1316 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6606  bren  6609  enssdom  6624
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