Theorem iunctb2

Description: Using the axiom of countable choice ax-cc , the countable union of countable sets is countable. See iunctb for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020)

	Ref	Expression
Assertion	iunctb2	⊢ ( ∀ 𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω )

Proof

Step	Hyp	Ref	Expression
1		omex	⊢ ω ∈ V
2		domrefg	⊢ ( ω ∈ V → ω ≼ ω )
3	1 2	ax-mp	⊢ ω ≼ ω
4		iunctb	⊢ ( ( ω ≼ ω ∧ ∀ 𝑥 ∈ ω 𝐵 ≼ ω ) → ∪ 𝑥 ∈ ω 𝐵 ≼ ω )
5	3 4	mpan	⊢ ( ∀ 𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω )