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Theorem bj-vprc 9889
Description: vprc 3885 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 9888 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2557 . . . . . . 7  |-  y  e. 
_V
32tbt 236 . . . . . 6  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1359 . . . . 5  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2034 . . . . 5  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 176 . . . 4  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1496 . . 3  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 595 . 2  |-  -.  E. x  x  =  _V
9 isset 2558 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
108, 9mtbir 596 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 98   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022  ax-bdn 9810  ax-bdel 9814  ax-bdsep 9877
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2556
This theorem is referenced by:  bj-nvel  9890  bj-vnex  9891  bj-intexr  9901  bj-intnexr  9902
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