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Theorem intnexr 3946
Description: If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intnexr  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )

Proof of Theorem intnexr
StepHypRef Expression
1 vprc 3929 . 2  |-  -.  _V  e.  _V
2 eleq1 2145 . 2  |-  ( |^| A  =  _V  ->  (
|^| A  e.  _V  <->  _V  e.  _V ) )
31, 2mtbiri 633 1  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    e. wcel 1434   _Vcvv 2610   |^|cint 3656
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 2065  ax-sep 3916
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612
This theorem is referenced by: (None)
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