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Metamath Proof Explorer

Theorem vprc

Description: The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of TakeutiZaring p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993) (Proof shortened by BJ, 1-May-2026)

		Ref	Expression
	Assertion	vprc	⊢ ¬ V ∈ V

Proof

Step	Hyp	Ref	Expression
1		nvel	⊢ ¬ V ∈ V