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	$$⊢\neg \mathrm{V}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		vnex	$$⊢\neg \exists xx=\mathrm{V}$$
2		isset	$$⊢\mathrm{V}\in \mathrm{V}\leftrightarrow \exists xx=\mathrm{V}$$
3	1 2	mtbir	$$⊢\neg \mathrm{V}\in \mathrm{V}$$