Axiom ax-cc 10447

Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 10489, but is weak enough that it can be proven using DC (see axcc 10470). 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 1539	. . 3 class 𝑥
3		com 7859	. . 3 class ω
4		cen 8954	. . 3 class ≈
5	2, 3, 4	wbr 5119	. 2 wff 𝑥 ≈ ω
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1539	. . . . . 6 class 𝑧
8		c0 4308	. . . . . 6 class ∅
9	7, 8	wne 2932	. . . . 5 wff 𝑧 ≠ ∅
10		vf	. . . . . . . 8 setvar 𝑓
11	10	cv 1539	. . . . . . 7 class 𝑓
12	7, 11	cfv 6530	. . . . . 6 class (𝑓‘𝑧)
13	12, 7	wcel 2108	. . . . 5 wff (𝑓‘𝑧) ∈ 𝑧
14	9, 13	wi 4	. . . 4 wff (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
15	14, 6, 2	wral 3051	. . 3 wff ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
16	15, 10	wex 1779	. 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  10448  axccdom  45194  axccd  45201