Theorem vprc 5267

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.)

Assertion

Ref	Expression
vprc	⊢ ¬ V ∈ V

Proof of Theorem vprc

Step	Hyp	Ref	Expression
1		nvel 5266	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2141  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455

This theorem is referenced by:  nvelOLD  5269  intex  5297  intnex  5298  abnex  7735  iprc  7887  opabn1stprc  8034  elfi2  9354  fi0  9360  ruALT  9551  cardmin2  9951  00lsp  21036  nowisdomv  30633  n0lplig  30643  fveqvfvv  47595  ndmaovcl  47758  vsn  49394  posnex  49562  prsnex  49563