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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