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Theorem vprc 4092
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 4091 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2743 . . . . . . 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 2250 . . . . 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 2744 . 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 2740
This theorem is referenced by:  nvel  4093  vnex  4094  intex  4109  intnex  4110  snnex  4461  iprc  4896  riotav  6242  elfi2  7101  fi0  7106  ruALT  7248  cardmin2  7564  00lsp  15665  inpc  24609
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 2237  ax-sep 4081
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-v 2742
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