Theorem vprc 5262

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 5261	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3456	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 323	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444

This theorem is referenced by:  nvel  5263  intex  5291  intnex  5292  abnex  7712  iprc  7863  opabn1stprc  8012  elfi2  9329  fi0  9335  ruALT  9523  cardmin2  9923  00lsp  20944  n0lplig  30571  fveqvfvv  47400  ndmaovcl  47563  vsn  49171  posnex  49339  prsnex  49340