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Theorem vprc 4562
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nalset 4561 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 3083 . . . . . . 7  |-  y  e. 
_V
32tbt 345 . . . . . 6  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1685 . . . . 5  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2415 . . . . 5  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 255 . . . 4  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1712 . . 3  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 299 . 2  |-  -.  E. x  x  =  _V
9 isset 3084 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
108, 9mtbir 300 1  |-  -.  _V  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3082
This theorem is referenced by:  nvel  4563  vnex  4564  intex  4580  intnex  4581  snnex  6611  iprc  6742  elfi2  7937  fi0  7943  ruALT  8125  cardmin2  8440  00lsp  18203  fveqvfvv  38496  ndmaovcl  38575
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