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

Metamath Proof Explorer

Theorem nalset

Description: No set contains all sets. Theorem 41 of Suppes p. 30. (Contributed by NM, 23-Aug-1993) Extract exnelv . (Revised by Matthew House, 12-Apr-2026)

		Ref	Expression
	Assertion	nalset	$⊢ \neg \exists x \forall y y \in x$

Proof

Step	Hyp	Ref	Expression
1		alexn	$⊢ \forall x \exists y \neg y \in x \leftrightarrow \neg \exists x \forall y y \in x$
2		exnelv	$⊢ \exists y \neg y \in x$
3	1 2	mpgbi	$⊢ \neg \exists x \forall y y \in x$