Metamath Proof Explorer

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	$⊢ \forall x \in ω B ≼ ω \to ⋃_{x \in ω} B ≼ ω$

Proof

Step	Hyp	Ref	Expression
1		omex	$⊢ ω \in V$
2		domrefg	$⊢ ω \in V \to ω ≼ ω$
3	1 2	ax-mp	$⊢ ω ≼ ω$
4		iunctb	$⊢ (ω ≼ ω \land \forall x \in ω B ≼ ω) \to ⋃_{x \in ω} B ≼ ω$
5	3 4	mpan	$⊢ \forall x \in ω B ≼ ω \to ⋃_{x \in ω} B ≼ ω$