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Axiom ax-cc 8061
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8104, but is weak enough that it can be proven using DC (see axcc 8084). 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 1622 . . 3  class  x
3 com 4656 . . 3  class  om
4 cen 6860 . . 3  class  ~~
52, 3, 4wbr 4023 . 2  wff  x  ~~  om
6 vz . . . . . . 7  set  z
76cv 1622 . . . . . 6  class  z
8 c0 3455 . . . . . 6  class  (/)
97, 8wne 2446 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  set  f
1110cv 1622 . . . . . . 7  class  f
127, 11cfv 5255 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1684 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2543 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1528 . 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  8062
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