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

Definition df-en 7601
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 7609. (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 7597 . 2  class  ~~
2 vx . . . . . 6  setvar  x
32cv 1454 . . . . 5  class  x
4 vy . . . . . 6  setvar  y
54cv 1454 . . . . 5  class  y
6 vf . . . . . 6  setvar  f
76cv 1454 . . . . 5  class  f
83, 5, 7wf1o 5604 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1674 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4476 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1455 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff setvar class
This definition is referenced by:  relen  7605  bren  7609  enssdom  7625
  Copyright terms: Public domain W3C validator