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

Theorem vprc 4960
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 4959 . 2 ¬ ∃𝑥 𝑥 = V
2 isset 3360 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
31, 2mtbir 314 1 ¬ V ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352
This theorem is referenced by:  nvel  4961  intex  4980  intnex  4981  abnex  7167  snnexOLD  7169  iprc  7303  opabn1stprc  7432  elfi2  8531  fi0  8537  ruALT  8719  cardmin2  9079  00lsp  19267  fveqvfvv  41841  ndmaovcl  41975
  Copyright terms: Public domain W3C validator