Theorem rexn0

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007)

	Ref	Expression
Assertion	rexn0	$$⊢\exists x\in A\phi \to A\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ne0i	$$⊢x\in A\to A\ne \varnothing$$
2	1	a1d	$$⊢x\in A\to \left(\phi \to A\ne \varnothing \right)$$
3	2	rexlimiv	$$⊢\exists x\in A\phi \to A\ne \varnothing$$