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Axiom ax-cc 9016
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9058, but is weak enough that it can be proven using DC (see axcc 9039). 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 1473 . . 3 class 𝑥
3 com 6833 . . 3 class ω
4 cen 7714 . . 3 class
52, 3, 4wbr 4481 . 2 wff 𝑥 ≈ ω
6 vz . . . . . . 7 setvar 𝑧
76cv 1473 . . . . . 6 class 𝑧
8 c0 3777 . . . . . 6 class
97, 8wne 2684 . . . . 5 wff 𝑧 ≠ ∅
10 vf . . . . . . . 8 setvar 𝑓
1110cv 1473 . . . . . . 7 class 𝑓
127, 11cfv 5689 . . . . . 6 class (𝑓𝑧)
1312, 7wcel 1938 . . . . 5 wff (𝑓𝑧) ∈ 𝑧
149, 13wi 4 . . . 4 wff (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1514, 6, 2wral 2800 . . 3 wff 𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1615, 10wex 1694 . 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  9017  axccdom  38309  axccd  38322
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