This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ φ \to \neg x \in A$
	Assertion	eq0rdvALT	$⊢ φ \to A = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		eq0rdvALT.1	$⊢ φ \to \neg x \in A$
2	1	pm2.21d	$⊢ φ \to (x \in A \to x \in \emptyset)$
3	2	ssrdv	$⊢ φ \to A \subseteq \emptyset$
4		ss0	$⊢ A \subseteq \emptyset \to A = \emptyset$
5	3 4	syl	$⊢ φ \to A = \emptyset$