Theorem eq0rdvALT

Description: Alternate proof of eq0rdv . Shorter, but requiring df-clel , ax-8 . (Contributed by NM, 11-Jul-2014) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	eq0rdvALT.1	$$⊢\phi \to \neg x\in A$$
Assertion	eq0rdvALT	$$⊢\phi \to A=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		eq0rdvALT.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	$$⊢\phi \to A=\varnothing$$