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Theorem intnexr 4166
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 4150 . 2  |-  -.  _V  e.  _V
2 eleq1 2252 . 2  |-  ( |^| A  =  _V  ->  (
|^| A  e.  _V  <->  _V  e.  _V ) )
31, 2mtbiri 676 1  |-  ( |^| A  =  _V  ->  -. 
|^| A  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364    e. wcel 2160   _Vcvv 2752   |^|cint 3859
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754
This theorem is referenced by: (None)
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