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Theorem vprc 4030
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  e.  _V

Proof of Theorem vprc
StepHypRef Expression
1 vnex 4029 . 2  |-  -.  E. x  x  =  _V
2 isset 2666 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
31, 2mtbir 645 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1316   E.wex 1453    e. wcel 1465   _Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  nvel  4031  intexr  4045  intnexr  4046  abnex  4338  snnex  4339  ruALT  4436  dcextest  4465  iprc  4777  snexxph  6806  elfi2  6828  fi0  6831
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