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

Axiom ax-cc 8728
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8770, but is weak enough that it can be proven using DC (see axcc 8751). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of nonempty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.)
Assertion
Ref Expression
ax-cc  |-  ( x 
~~  om  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
Distinct variable group:    x, f, z

Detailed syntax breakdown of Axiom ax-cc
StepHypRef Expression
1 vx . . . 4  setvar  x
21cv 1398 . . 3  class  x
3 com 6599 . . 3  class  om
4 cen 7432 . . 3  class  ~~
52, 3, 4wbr 4367 . 2  wff  x  ~~  om
6 vz . . . . . . 7  setvar  z
76cv 1398 . . . . . 6  class  z
8 c0 3711 . . . . . 6  class  (/)
97, 8wne 2577 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  setvar  f
1110cv 1398 . . . . . . 7  class  f
127, 11cfv 5496 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1826 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2732 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1620 . 2  wff  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z )
175, 16wi 4 1  wff  ( x 
~~  om  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
Colors of variables: wff setvar class
This axiom is referenced by:  axcc2lem  8729
  Copyright terms: Public domain W3C validator