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Axiom ax-cc 10475
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10517, but is weak enough that it can be proven using DC (see axcc 10498). 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 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
Distinct variable group:   𝑥,𝑓,𝑧

Detailed syntax breakdown of Axiom ax-cc
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
21cv 1539 . . 3 class 𝑥
3 com 7887 . . 3 class ω
4 cen 8982 . . 3 class
52, 3, 4wbr 5143 . 2 wff 𝑥 ≈ ω
6 vz . . . . . . 7 setvar 𝑧
76cv 1539 . . . . . 6 class 𝑧
8 c0 4333 . . . . . 6 class
97, 8wne 2940 . . . . 5 wff 𝑧 ≠ ∅
10 vf . . . . . . . 8 setvar 𝑓
1110cv 1539 . . . . . . 7 class 𝑓
127, 11cfv 6561 . . . . . 6 class (𝑓𝑧)
1312, 7wcel 2108 . . . . 5 wff (𝑓𝑧) ∈ 𝑧
149, 13wi 4 . . . 4 wff (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1514, 6, 2wral 3061 . . 3 wff 𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1615, 10wex 1779 . 2 wff 𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
175, 16wi 4 1 wff (𝑥 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
Colors of variables: wff setvar class
This axiom is referenced by:  axcc2lem  10476  axccdom  45227  axccd  45234
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