Theorem vprc 5316

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.)

Assertion

Ref	Expression
vprc	⊢ ¬ V ∈ V

Proof of Theorem vprc

Step	Hyp	Ref	Expression
1		vnex 5315	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3488	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 323	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477

This theorem is referenced by:  nvel  5317  intex  5338  intnex  5339  abnex  7744  iprc  7904  opabn1stprc  8044  elfi2  9409  fi0  9415  ruALT  9598  cardmin2  9994  00lsp  20592  n0lplig  29736  fveqvfvv  45750  ndmaovcl  45911  vsn  47496