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Metamath Proof Explorer

Theorem nvel

Description: The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006) Prove it without using vprc , which is then proved as an instance of it. (Revised by BJ, 1-May-2026)

		Ref	Expression
	Assertion	nvel	⊢ ¬ V ∈ 𝐴

Proof

Step	Hyp	Ref	Expression
1		vnex	⊢ ¬ ∃ 𝑥 𝑥 = V
2		elisset	⊢ ( V ∈ 𝐴 → ∃ 𝑥 𝑥 = V )
3	1 2	mto	⊢ ¬ V ∈ 𝐴