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Theorem nvelim 27978
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4153. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nvelim  |-  ( A  =  _V  ->  -.  A  e.  B )

Proof of Theorem nvelim
StepHypRef Expression
1 nvel 4153 . 2  |-  -.  _V  e.  B
2 eleq1 2343 . . 3  |-  ( _V  =  A  ->  ( _V  e.  B  <->  A  e.  B ) )
32eqcoms 2286 . 2  |-  ( A  =  _V  ->  ( _V  e.  B  <->  A  e.  B ) )
41, 3mtbii 293 1  |-  ( A  =  _V  ->  -.  A  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  afvvdm  28004  afvvfunressn  28006  afvvv  28008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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