Theorem unictb

Description: The countable union of countable sets is countable. Theorem 6Q of Enderton p. 159. See iunctb for indexed union version. (Contributed by NM, 26-Mar-2006)

	Ref	Expression
Assertion	unictb	$$⊢\left(A\preccurlyeq \mathrm{\omega }\wedge \forall x\in Ax\preccurlyeq \mathrm{\omega }\right)\to \bigcup A\preccurlyeq \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		uniiun	$$⊢\bigcup A=\bigcup _{x\in A}x$$
2		iunctb	$$⊢\left(A\preccurlyeq \mathrm{\omega }\wedge \forall x\in Ax\preccurlyeq \mathrm{\omega }\right)\to \bigcup _{x\in A}x\preccurlyeq \mathrm{\omega }$$
3	1 2	eqbrtrid	$$⊢\left(A\preccurlyeq \mathrm{\omega }\wedge \forall x\in Ax\preccurlyeq \mathrm{\omega }\right)\to \bigcup A\preccurlyeq \mathrm{\omega }$$