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Theorem vprc 4049
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 nalset 4048 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2730 . . . . . . 7  |-  y  e. 
_V
32tbt 335 . . . . . 6  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1554 . . . . 5  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2247 . . . . 5  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 245 . . . 4  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1580 . . 3  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 291 . 2  |-  -.  E. x  x  =  _V
9 isset 2731 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
108, 9mtbir 292 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727
This theorem is referenced by:  nvel  4050  vnex  4051  intex  4065  intnex  4066  snnex  4415  iprc  4850  riotav  6195  elfi2  7052  fi0  7057  ruALT  7199  cardmin2  7515  00lsp  15573  inpc  24443
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-8 1623  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-9 1684  ax-4 1692  ax-ext 2234  ax-sep 4038
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-v 2729
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