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Axiom ax-cc 9594
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9636, but is weak enough that it can be proven using DC (see axcc 9617). 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 1600 . . 3 class 𝑥
3 com 7345 . . 3 class ω
4 cen 8240 . . 3 class
52, 3, 4wbr 4888 . 2 wff 𝑥 ≈ ω
6 vz . . . . . . 7 setvar 𝑧
76cv 1600 . . . . . 6 class 𝑧
8 c0 4141 . . . . . 6 class
97, 8wne 2969 . . . . 5 wff 𝑧 ≠ ∅
10 vf . . . . . . . 8 setvar 𝑓
1110cv 1600 . . . . . . 7 class 𝑓
127, 11cfv 6137 . . . . . 6 class (𝑓𝑧)
1312, 7wcel 2107 . . . . 5 wff (𝑓𝑧) ∈ 𝑧
149, 13wi 4 . . . 4 wff (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1514, 6, 2wral 3090 . . 3 wff 𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
1615, 10wex 1823 . 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  9595  axccdom  40351  axccd  40360
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