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Theorem bj-vprc 10845
Description: vprc 3917 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-vprc  |-  -.  _V  e.  _V

Proof of Theorem bj-vprc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-nalset 10844 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2605 . . . . . . 7  |-  y  e. 
_V
32tbt 245 . . . . . 6  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1400 . . . . 5  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2076 . . . . 5  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 185 . . . 4  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1537 . . 3  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 628 . 2  |-  -.  E. x  x  =  _V
9 isset 2606 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
108, 9mtbir 629 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602
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 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064  ax-bdn 10766  ax-bdel 10770  ax-bdsep 10833
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by:  bj-nvel  10846  bj-vnex  10847  bj-intexr  10857  bj-intnexr  10858
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