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