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

Definition df-ac 7676
Description: The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 8018 as our definition, because the equivalence to more standard forms (dfac2 7690) requires the Axiom of Regularity, which we often try to avoid. Thus we take the first of the "textbook forms" as the definition and derive the form of ax-ac 8018 itself as dfac0 7692. (Contributed by Mario Carneiro, 22-Feb-2015.)

Assertion
Ref Expression
df-ac  |-  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x ) )
Distinct variable group:    x, f

Detailed syntax breakdown of Definition df-ac
StepHypRef Expression
1 wac 7675 . 2  wff CHOICE
2 vf . . . . . . 7  set  f
32cv 1618 . . . . . 6  class  f
4 vx . . . . . . 7  set  x
54cv 1618 . . . . . 6  class  x
63, 5wss 3094 . . . . 5  wff  f  C_  x
75cdm 4626 . . . . . 6  class  dom  x
83, 7wfn 4633 . . . . 5  wff  f  Fn 
dom  x
96, 8wa 360 . . . 4  wff  ( f 
C_  x  /\  f  Fn  dom  x )
109, 2wex 1537 . . 3  wff  E. f
( f  C_  x  /\  f  Fn  dom  x )
1110, 4wal 1532 . 2  wff  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
121, 11wb 178 1  wff  (CHOICE  <->  A. x E. f ( f  C_  x  /\  f  Fn  dom  x ) )
Colors of variables: wff set class
This definition is referenced by:  dfac3  7681  ac7  8033
  Copyright terms: Public domain W3C validator