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	|- ( A. x e. _om B ~<_ _om -> U_ x e. _om B ~<_ _om )

Proof

Step	Hyp	Ref	Expression
1		omex	|- _om e. _V
2		domrefg	|- ( _om e. _V -> _om ~<_ _om )
3	1 2	ax-mp	|- _om ~<_ _om
4		iunctb	|- ( ( _om ~<_ _om /\ A. x e. _om B ~<_ _om ) -> U_ x e. _om B ~<_ _om )
5	3 4	mpan	|- ( A. x e. _om B ~<_ _om -> U_ x e. _om B ~<_ _om )