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Axiom ax-cc 8077
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8120, but is weak enough that it can be proven using DC (see axcc 8100). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of non-empty 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  set  x
21cv 1631 . . 3  class  x
3 com 4672 . . 3  class  om
4 cen 6876 . . 3  class  ~~
52, 3, 4wbr 4039 . 2  wff  x  ~~  om
6 vz . . . . . . 7  set  z
76cv 1631 . . . . . 6  class  z
8 c0 3468 . . . . . 6  class  (/)
97, 8wne 2459 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  set  f
1110cv 1631 . . . . . . 7  class  f
127, 11cfv 5271 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1696 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2556 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1531 . 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 set class
This axiom is referenced by:  axcc2lem  8078
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