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Theorem vnex 4059
Description: The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
Assertion
Ref Expression
vnex  |-  -.  E. x  x  =  _V

Proof of Theorem vnex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nalset 4058 . 2  |-  -.  E. x A. y  y  e.  x
2 vex 2689 . . . . . 6  |-  y  e. 
_V
32tbt 246 . . . . 5  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1446 . . . 4  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2133 . . . 4  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 186 . . 3  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1584 . 2  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 659 1  |-  -.  E. x  x  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121  ax-sep 4046
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688
This theorem is referenced by:  vprc  4060
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