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Theorem vprc 4126
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 4125 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2766 . . . . . . 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 2252 . . . . 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 2767 . 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 2763
This theorem is referenced by:  nvel  4127  vnex  4128  intex  4143  intnex  4144  snnex  4496  iprc  4931  riotav  6277  elfi2  7136  fi0  7141  ruALT  7283  cardmin2  7599  00lsp  15701  inpc  24645
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 2239  ax-sep 4115
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 2245  df-cleq 2251  df-clel 2254  df-v 2765
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