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Axiom ax-cc 9449
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9491, but is weak enough that it can be proven using DC (see axcc 9472). 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 1631 . . 3 class 𝑥
3 com 7230 . . 3 class ω
4 cen 8118 . . 3 class
52, 3, 4wbr 4804 . 2 wff 𝑥 ≈ ω
6 vz . . . . . . 7 setvar 𝑧
76cv 1631 . . . . . 6 class 𝑧
8 c0 4058 . . . . . 6 class
97, 8wne 2932 . . . . 5 wff 𝑧 ≠ ∅
10 vf . . . . . . . 8 setvar 𝑓
1110cv 1631 . . . . . . 7 class 𝑓
127, 11cfv 6049 . . . . . 6 class (𝑓𝑧)
1312, 7wcel 2139 . . . . 5 wff (𝑓𝑧) ∈ 𝑧
149, 13wi 4 . . . 4 wff (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1514, 6, 2wral 3050 . . 3 wff 𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1615, 10wex 1853 . 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  9450  axccdom  39915  axccd  39928
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