Theorem ssnum

Description: A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	ssnum	$$⊢\left(A\in \mathrm{dom}\mathrm{card}\wedge B\subseteq A\right)\to B\in \mathrm{dom}\mathrm{card}$$

Proof

Step	Hyp	Ref	Expression
1		ssdomg	$$⊢A\in \mathrm{dom}\mathrm{card}\to \left(B\subseteq A\to B\preccurlyeq A\right)$$
2	1	imp	$$⊢\left(A\in \mathrm{dom}\mathrm{card}\wedge B\subseteq A\right)\to B\preccurlyeq A$$
3		numdom	$$⊢\left(A\in \mathrm{dom}\mathrm{card}\wedge B\preccurlyeq A\right)\to B\in \mathrm{dom}\mathrm{card}$$
4	2 3	syldan	$$⊢\left(A\in \mathrm{dom}\mathrm{card}\wedge B\subseteq A\right)\to B\in \mathrm{dom}\mathrm{card}$$