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Theorem vnex 3962
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 3961 . 2  |-  -.  E. x A. y  y  e.  x
2 vex 2622 . . . . . 6  |-  y  e. 
_V
32tbt 245 . . . . 5  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1404 . . . 4  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2082 . . . 4  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 185 . . 3  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1541 . 2  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 630 1  |-  -.  E. x  x  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  vprc  3963
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