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

Axiom ax-cc 8883
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8925, but is weak enough that it can be proven using DC (see axcc 8906). 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 1451 . . 3  class  x
3 com 6711 . . 3  class  om
4 cen 7584 . . 3  class  ~~
52, 3, 4wbr 4395 . 2  wff  x  ~~  om
6 vz . . . . . . 7  setvar  z
76cv 1451 . . . . . 6  class  z
8 c0 3722 . . . . . 6  class  (/)
97, 8wne 2641 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  setvar  f
1110cv 1451 . . . . . . 7  class  f
127, 11cfv 5589 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1904 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2756 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1671 . 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  8884
  Copyright terms: Public domain W3C validator