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Definition df-en 6253
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 6259. (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 6250 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1258 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1258 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1258 . . . . 5  class  f
83, 5, 7wf1o 4929 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1397 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 3845 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1259 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6256  bren  6259  enssdom  6273
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