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 → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))

Distinct variable group:   𝑥,𝑓,𝑧

Detailed syntax breakdown of Axiom ax-cc

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2	1	cv 1600	. . 3 class 𝑥
3		com 7345	. . 3 class ω
4		cen 8240	. . 3 class ≈
5	2, 3, 4	wbr 4888	. 2 wff 𝑥 ≈ ω
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1600	. . . . . 6 class 𝑧
8		c0 4141	. . . . . 6 class ∅
9	7, 8	wne 2969	. . . . 5 wff 𝑧 ≠ ∅
10		vf	. . . . . . . 8 setvar 𝑓
11	10	cv 1600	. . . . . . 7 class 𝑓
12	7, 11	cfv 6137	. . . . . 6 class (𝑓‘𝑧)
13	12, 7	wcel 2107	. . . . 5 wff (𝑓‘𝑧) ∈ 𝑧
14	9, 13	wi 4	. . . 4 wff (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
15	14, 6, 2	wral 3090	. . 3 wff ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
16	15, 10	wex 1823	. 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))

This axiom is referenced by:  axcc2lem  9595  axccdom  40351  axccd  40360