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Axiom ax-cc 9217
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9259, but is weak enough that it can be proven using DC (see axcc 9240). 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 1479 . . 3 class 𝑥
3 com 7027 . . 3 class ω
4 cen 7912 . . 3 class
52, 3, 4wbr 4623 . 2 wff 𝑥 ≈ ω
6 vz . . . . . . 7 setvar 𝑧
76cv 1479 . . . . . 6 class 𝑧
8 c0 3897 . . . . . 6 class
97, 8wne 2790 . . . . 5 wff 𝑧 ≠ ∅
10 vf . . . . . . . 8 setvar 𝑓
1110cv 1479 . . . . . . 7 class 𝑓
127, 11cfv 5857 . . . . . 6 class (𝑓𝑧)
1312, 7wcel 1987 . . . . 5 wff (𝑓𝑧) ∈ 𝑧
149, 13wi 4 . . . 4 wff (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1514, 6, 2wral 2908 . . 3 wff 𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1615, 10wex 1701 . 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  9218  axccdom  38925  axccd  38938
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