Theorem vprc 5183

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 5182	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3453	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 326	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5167

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443

This theorem is referenced by:  nvel  5184  intex  5204  intnex  5205  abnex  7459  iprc  7600  opabn1stprc  7738  elfi2  8862  fi0  8868  ruALT  9051  cardmin2  9412  00lsp  19746  n0lplig  28266  fveqvfvv  43632  ndmaovcl  43759