MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vprc Structured version   Visualization version   GIF version

Theorem vprc 5192
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
StepHypRef Expression
1 vnex 5191 . 2 ¬ ∃𝑥 𝑥 = V
2 isset 3483 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
31, 2mtbir 326 1 ¬ V ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wex 1781  wcel 2115  Vcvv 3471
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793  ax-sep 5176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473
This theorem is referenced by:  nvel  5193  intex  5213  intnex  5214  abnex  7454  iprc  7593  opabn1stprc  7731  elfi2  8854  fi0  8860  ruALT  9043  cardmin2  9404  00lsp  19728  n0lplig  28244  fveqvfvv  43425  ndmaovcl  43552
  Copyright terms: Public domain W3C validator