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Theorem vprc 3858
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
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nalset 3857 . . 3
2 vex 2534 . . . . . . 7 
_V
32tbt 236 . . . . . 6 
_V
43albii 1335 . . . . 5  _V
5 dfcleq 2012 . . . . 5  _V  _V
64, 5bitr4i 176 . . . 4  _V
76exbii 1474 . . 3  _V
81, 7mtbi 582 . 2  _V
9 isset 2535 . 2  _V  _V  _V
108, 9mtbir 583 1  _V  _V
Colors of variables: wff set class
Syntax hints:   wn 3   wb 98  wal 1224   wceq 1226  wex 1358   wcel 1370   _Vcvv 2531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000  ax-sep 3845
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533
This theorem is referenced by:  nvel  3859  vnex  3860  intexr  3874  intnexr  3875  snnex  4127  ruALT  4209  iprc  4523
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