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Theorem vnex 4146
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 4145 . 2  |-  -.  E. x A. y  y  e.  x
2 vex 2752 . . . . . 6  |-  y  e. 
_V
32tbt 247 . . . . 5  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1480 . . . 4  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2181 . . . 4  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 187 . . 3  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1615 . 2  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 671 1  |-  -.  E. x  x  =  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 105   A.wal 1361    = wceq 1363   E.wex 1502    e. wcel 2158   _Vcvv 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751
This theorem is referenced by:  vprc  4147
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