Axiom ax-cc 10389

Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10431, but is weak enough that it can be proven using DC (see axcc 10412). 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 1558	. . 3 class 𝑥
3		com 7842	. . 3 class ω
4		cen 8920	. . 3 class ≈
5	2, 3, 4	wbr 5099	. 2 wff 𝑥 ≈ ω
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1558	. . . . . 6 class 𝑧
8		c0 4285	. . . . . 6 class ∅
9	7, 8	wne 2956	. . . . 5 wff 𝑧 ≠ ∅
10		vf	. . . . . . . 8 setvar 𝑓
11	10	cv 1558	. . . . . . 7 class 𝑓
12	7, 11	cfv 6517	. . . . . 6 class (𝑓‘𝑧)
13	12, 7	wcel 2141	. . . . 5 wff (𝑓‘𝑧) ∈ 𝑧
14	9, 13	wi 4	. . . 4 wff (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
15	14, 6, 2	wral 3075	. . 3 wff ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
16	15, 10	wex 1798	. 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  10390  axccdom  45762  axccd  45768