Axiom ax-cc 10517

Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10559, but is weak enough that it can be proven using DC (see axcc 10540). 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 1568	. . 3 class 𝑥
3		com 7878	. . 3 class ω
4		cen 8974	. . 3 class ≈
5	2, 3, 4	wbr 5102	. 2 wff 𝑥 ≈ ω
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1568	. . . . . 6 class 𝑧
8		c0 4278	. . . . . 6 class ∅
9	7, 8	wne 2955	. . . . 5 wff 𝑧 ≠ ∅
10		vf	. . . . . . . 8 setvar 𝑓
11	10	cv 1568	. . . . . . 7 class 𝑓
12	7, 11	cfv 6539	. . . . . 6 class (𝑓‘𝑧)
13	12, 7	wcel 2144	. . . . 5 wff (𝑓‘𝑧) ∈ 𝑧
14	9, 13	wi 4	. . . 4 wff (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
15	14, 6, 2	wral 3076	. . 3 wff ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
16	15, 10	wex 1811	. 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 used by:  axcc2lem  10518  axccdom  46329  axccd  46335