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

Definition df-en 7101
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 7108. (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 7097 . 2  class  ~~
2 vx . . . . . 6  set  x
32cv 1651 . . . . 5  class  x
4 vy . . . . . 6  set  y
54cv 1651 . . . . 5  class  y
6 vf . . . . . 6  set  f
76cv 1651 . . . . 5  class  f
83, 5, 7wf1o 5444 . . . 4  wff  f : x -1-1-onto-> y
98, 6wex 1550 . . 3  wff  E. f 
f : x -1-1-onto-> y
109, 2, 4copab 4257 . 2  class  { <. x ,  y >.  |  E. f  f : x -1-1-onto-> y }
111, 10wceq 1652 1  wff  ~~  =  { <. x ,  y
>.  |  E. f 
f : x -1-1-onto-> y }
Colors of variables: wff set class
This definition is referenced by:  relen  7105  bren  7108  enssdom  7123
  Copyright terms: Public domain W3C validator