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Axiom ax-cc 8872
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8914, but is weak enough that it can be proven using DC (see axcc 8895). 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 1436 . . 3  class  x
3 com 6706 . . 3  class  om
4 cen 7577 . . 3  class  ~~
52, 3, 4wbr 4423 . 2  wff  x  ~~  om
6 vz . . . . . . 7  setvar  z
76cv 1436 . . . . . 6  class  z
8 c0 3761 . . . . . 6  class  (/)
97, 8wne 2614 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  setvar  f
1110cv 1436 . . . . . . 7  class  f
127, 11cfv 5601 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1872 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2771 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1657 . 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  8873
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