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Axiom ax-cc 8250
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8292, but is weak enough that it can be proven using DC (see axcc 8273). 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 1648 . . 3  class  x
3 com 4787 . . 3  class  om
4 cen 7044 . . 3  class  ~~
52, 3, 4wbr 4155 . 2  wff  x  ~~  om
6 vz . . . . . . 7  set  z
76cv 1648 . . . . . 6  class  z
8 c0 3573 . . . . . 6  class  (/)
97, 8wne 2552 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  set  f
1110cv 1648 . . . . . . 7  class  f
127, 11cfv 5396 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1717 . . . . 5  wff  ( f `
 z )  e.  z
149, 13wi 4 . . . 4  wff  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1514, 6, 2wral 2651 . . 3  wff  A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )
1615, 10wex 1547 . 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  8251
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