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Axiom ax-cc 8592
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8634, but is weak enough that it can be proven using DC (see axcc 8615). 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 1361 . . 3  class  x
3 com 6465 . . 3  class  om
4 cen 7295 . . 3  class  ~~
52, 3, 4wbr 4280 . 2  wff  x  ~~  om
6 vz . . . . . . 7  setvar  z
76cv 1361 . . . . . 6  class  z
8 c0 3625 . . . . . 6  class  (/)
97, 8wne 2596 . . . . 5  wff  z  =/=  (/)
10 vf . . . . . . . 8  setvar  f
1110cv 1361 . . . . . . 7  class  f
127, 11cfv 5406 . . . . . 6  class  ( f `
 z )
1312, 7wcel 1755 . . . . 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 1589 . 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  8593
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