Theorem eq0rdv

Description: Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014)

	Ref	Expression
Hypothesis	eq0rdv.1	$$⊢\phi \to \neg x\in A$$
Assertion	eq0rdv	$$⊢\phi \to A=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		eq0rdv.1	$$⊢\phi \to \neg x\in A$$
2	1	pm2.21d	$$⊢\phi \to \left(x\in A\to x\in \varnothing \right)$$
3	2	ssrdv	$$⊢\phi \to A\subseteq \varnothing$$
4		ss0	$$⊢A\subseteq \varnothing \to A=\varnothing$$
5	3 4	syl	$$⊢\phi \to A=\varnothing$$