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Definition df-en 6860
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 6867. (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 6856 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1622 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1622 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1622 . . . . 5  class  f
83, 5, 7wf1o 5220 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1528 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4077 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1623 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6864  bren  6867  enssdom  6882
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