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

Definition df-ac 7990
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 8332 as our definition, because the equivalence to more standard forms (dfac2 8004) 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 8332 itself as dfac0 8006. (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 7989 . 2  wff CHOICE
2 vf . . . . . . 7  set  f
32cv 1651 . . . . . 6  class  f
4 vx . . . . . . 7  set  x
54cv 1651 . . . . . 6  class  x
63, 5wss 3313 . . . . 5  wff  f  C_  x
75cdm 4871 . . . . . 6  class  dom  x
83, 7wfn 5442 . . . . 5  wff  f  Fn 
dom  x
96, 8wa 359 . . . 4  wff  ( f 
C_  x  /\  f  Fn  dom  x )
109, 2wex 1550 . . 3  wff  E. f
( f  C_  x  /\  f  Fn  dom  x )
1110, 4wal 1549 . 2  wff  A. x E. f ( f  C_  x  /\  f  Fn  dom  x )
121, 11wb 177 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  7995  ac7  8346
  Copyright terms: Public domain W3C validator