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

Metamath Proof Explorer

Theorem nel0

Description: From the general negation of membership in A , infer that A is the empty set. (Contributed by BJ, 6-Oct-2018)

		Ref	Expression
	Hypothesis	nel0.1	$⊢ \neg x \in A$
	Assertion	nel0	$⊢ A = \emptyset$

Step	Hyp	Ref	Expression
1		nel0.1	$⊢ \neg x \in A$
2		eq0	$⊢ A = \emptyset \leftrightarrow \forall x \neg x \in A$
3	2 1	mpgbir	$⊢ A = \emptyset$