Theorem ssct

Description: Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017)

	Ref	Expression
Assertion	ssct	$$⊢\left(A\subseteq B\wedge B\preccurlyeq \mathrm{\omega }\right)\to A\preccurlyeq \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		ctex	$$⊢B\preccurlyeq \mathrm{\omega }\to B\in \mathrm{V}$$
2		ssdomg	$$⊢B\in \mathrm{V}\to \left(A\subseteq B\to A\preccurlyeq B\right)$$
3	1 2	syl	$$⊢B\preccurlyeq \mathrm{\omega }\to \left(A\subseteq B\to A\preccurlyeq B\right)$$
4	3	impcom	$$⊢\left(A\subseteq B\wedge B\preccurlyeq \mathrm{\omega }\right)\to A\preccurlyeq B$$
5		domtr	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq \mathrm{\omega }\right)\to A\preccurlyeq \mathrm{\omega }$$
6	4 5	sylancom	$$⊢\left(A\subseteq B\wedge B\preccurlyeq \mathrm{\omega }\right)\to A\preccurlyeq \mathrm{\omega }$$