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

Definition df-en 6880
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 6887. (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 6876 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1631 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1631 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1631 . . . . 5  class  f
83, 5, 7wf1o 5270 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1531 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4092 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1632 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  6884  bren  6887  enssdom  6902
  Copyright terms: Public domain W3C validator