Axiom ax-cc

Description: The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 , but is weak enough that it can be proven using DC (see axcc ). 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)

	Ref	Expression
Assertion	ax-cc	⊢ ( 𝑥 ≈ ω → ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vx	⊢ 𝑥
1	0	cv	⊢ 𝑥
2		cen	⊢ ≈
3		com	⊢ ω
4	1 3 2	wbr	⊢ 𝑥 ≈ ω
5		vf	⊢ 𝑓
6		vz	⊢ 𝑧
7	6	cv	⊢ 𝑧
8		c0	⊢ ∅
9	7 8	wne	⊢ 𝑧 ≠ ∅
10	5	cv	⊢ 𝑓
11	7 10	cfv	⊢ ( 𝑓 ‘ 𝑧 )
12	11 7	wcel	⊢ ( 𝑓 ‘ 𝑧 ) ∈ 𝑧
13	9 12	wi	⊢ ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 )
14	13 6 1	wral	⊢ ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 )
15	14 5	wex	⊢ ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 )
16	4 15	wi	⊢ ( 𝑥 ≈ ω → ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) )