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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)

		Ref	Expression
	Assertion	vprc	⊢ ¬ V ∈ V

Proof

Step	Hyp	Ref	Expression
1		vnex	⊢ ¬ ∃ 𝑥 𝑥 = V
2		isset	⊢ ( V ∈ V ↔ ∃ 𝑥 𝑥 = V )
3	1 2	mtbir	⊢ ¬ V ∈ V