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Axiom ax-cc 8057
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8100, but is weak enough that it can be proven using DC (see axcc 8080). 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 1623 . . 3  class  x
3 com 4656 . . 3  class  om
4 cen 6856 . . 3  class  ~~
52, 3, 4wbr 4025 . 2  wff  x  ~~  om
6 vz . . . . . . 7  set  z
76cv 1623 . . . . . 6  class  z
8 c0 3457 . . . . . 6  class  (/)
97, 8wne 2448 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  set  f
1110cv 1623 . . . . . . 7  class  f
127, 11cfv 5222 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1685 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 6 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2545 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1529 . 2  wff  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z )
175, 16wi 6 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  8058
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