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Theorem vprc 3916
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 3915 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2577 . . . . . . 7  |-  y  e. 
_V
32tbt 240 . . . . . 6  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1375 . . . . 5  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2050 . . . . 5  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 180 . . . 4  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1512 . . 3  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 605 . 2  |-  -.  E. x  x  =  _V
9 isset 2578 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
108, 9mtbir 606 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  nvel  3917  vnex  3918  intexr  3932  intnexr  3933  snnex  4209  ruALT  4303  iprc  4628
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