Theorem ssn0rex

Description: There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020)

	Ref	Expression
Assertion	ssn0rex	$$⊢\left(A\subseteq B\wedge A\ne \varnothing \right)\to \exists x\in Bx\in A$$

Proof

Step	Hyp	Ref	Expression
1		ssrexv	$$⊢A\subseteq B\to \left(\exists x\in Ax\in A\to \exists x\in Bx\in A\right)$$
2		n0rex	$$⊢A\ne \varnothing \to \exists x\in Ax\in A$$
3	1 2	impel	$$⊢\left(A\subseteq B\wedge A\ne \varnothing \right)\to \exists x\in Bx\in A$$