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Mirrors > Home > MPE Home > Th. List > ax-cc | Structured version Visualization version GIF version |
Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9899, but is weak enough that it can be proven using DC (see axcc 9880). 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.) |
Ref | Expression |
---|---|
ax-cc | ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . 4 setvar 𝑥 | |
2 | 1 | cv 1536 | . . 3 class 𝑥 |
3 | com 7580 | . . 3 class ω | |
4 | cen 8506 | . . 3 class ≈ | |
5 | 2, 3, 4 | wbr 5066 | . 2 wff 𝑥 ≈ ω |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1536 | . . . . . 6 class 𝑧 |
8 | c0 4291 | . . . . . 6 class ∅ | |
9 | 7, 8 | wne 3016 | . . . . 5 wff 𝑧 ≠ ∅ |
10 | vf | . . . . . . . 8 setvar 𝑓 | |
11 | 10 | cv 1536 | . . . . . . 7 class 𝑓 |
12 | 7, 11 | cfv 6355 | . . . . . 6 class (𝑓‘𝑧) |
13 | 12, 7 | wcel 2114 | . . . . 5 wff (𝑓‘𝑧) ∈ 𝑧 |
14 | 9, 13 | wi 4 | . . . 4 wff (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) |
15 | 14, 6, 2 | wral 3138 | . . 3 wff ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) |
16 | 15, 10 | wex 1780 | . 2 wff ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) |
17 | 5, 16 | wi 4 | 1 wff (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) |
Colors of variables: wff setvar class |
This axiom is referenced by: axcc2lem 9858 axccdom 41536 axccd 41544 |
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