Axiom ax-cc 10184

Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10226, but is weak enough that it can be proven using DC (see axcc 10207). 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 1541	. . 3 class 𝑥
3		com 7701	. . 3 class ω
4		cen 8705	. . 3 class ≈
5	2, 3, 4	wbr 5079	. 2 wff 𝑥 ≈ ω
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		c0 4262	. . . . . 6 class ∅
9	7, 8	wne 2945	. . . . 5 wff 𝑧 ≠ ∅
10		vf	. . . . . . . 8 setvar 𝑓
11	10	cv 1541	. . . . . . 7 class 𝑓
12	7, 11	cfv 6431	. . . . . 6 class (𝑓‘𝑧)
13	12, 7	wcel 2110	. . . . 5 wff (𝑓‘𝑧) ∈ 𝑧
14	9, 13	wi 4	. . . 4 wff (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
15	14, 6, 2	wral 3066	. . 3 wff ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
16	15, 10	wex 1786	. 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  10185  axccdom  42724  axccd  42730