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Axiom ax-cc 8315
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8357, but is weak enough that it can be proven using DC (see axcc 8338). 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 1651 . . 3  class  x
3 com 4845 . . 3  class  om
4 cen 7106 . . 3  class  ~~
52, 3, 4wbr 4212 . 2  wff  x  ~~  om
6 vz . . . . . . 7  set  z
76cv 1651 . . . . . 6  class  z
8 c0 3628 . . . . . 6  class  (/)
97, 8wne 2599 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  set  f
1110cv 1651 . . . . . . 7  class  f
127, 11cfv 5454 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1725 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2705 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1550 . 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  8316
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