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Metamath Proof Explorer

Theorem vneqv

Description: The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005) Extract from vnex and shorten proof. (Revised by BJ, 25-Apr-2026)

		Ref	Expression
	Assertion	vneqv	$⊢ \neg x = V$

Proof

Step	Hyp	Ref	Expression
1		vex	$⊢ y \in V$
2		eleq2	$⊢ x = V \to (y \in x \leftrightarrow y \in V)$
3	1 2	mpbiri	$⊢ x = V \to y \in x$
4	3	con3i	$⊢ \neg y \in x \to \neg x = V$
5		exnelv	$⊢ \exists y \neg y \in x$
6	4 5	exlimiiv	$⊢ \neg x = V$