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Definition df-en 6750
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 6757. (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 6746 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1618 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1618 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1618 . . . . 5  class  f
83, 5, 7wf1o 4591 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1537 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 3973 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1619 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6754  bren  6757  enssdom  6772
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