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Axiom ax-cc 8815
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8857, but is weak enough that it can be proven using DC (see axcc 8838). 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 1378 . . 3  class  x
3 com 6684 . . 3  class  om
4 cen 7513 . . 3  class  ~~
52, 3, 4wbr 4447 . 2  wff  x  ~~  om
6 vz . . . . . . 7  setvar  z
76cv 1378 . . . . . 6  class  z
8 c0 3785 . . . . . 6  class  (/)
97, 8wne 2662 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  setvar  f
1110cv 1378 . . . . . . 7  class  f
127, 11cfv 5588 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1767 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2814 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1596 . 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  8816
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