Theorem vprc 5281

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 5279	1 ⊢ ¬ V ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by:  nvelOLD  5283  intex  5312  intnex  5313  abnex  7803  iprc  7957  opabn1stprc  8104  elfi2  9443  fi0  9449  ruALT  9640  cardmin2  10119  00lsp  21307  nowisdomv  31132  n0lplig  31142  fveqvfvv  48383  ndmaovcl  48546  vsn  50254  posnex  50425  prsnex  50426