MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-en Unicode version

Definition df-en 6797
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 6804. (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 6793 . 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 4637 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1537 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4016 . 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  6801  bren  6804  enssdom  6819
  Copyright terms: Public domain W3C validator