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Theorem vprc 3963
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 3962 . 2  |-  -.  E. x  x  =  _V
2 isset 2625 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
31, 2mtbir 631 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  nvel  3964  intexr  3978  intnexr  3979  snnex  4262  ruALT  4357  dcextest  4386  iprc  4689  snexxph  6638
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