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Theorem nvelim 27845
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4302. (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 4302 . 2  |-  -.  _V  e.  B
2 eleq1 2464 . . 3  |-  ( _V  =  A  ->  ( _V  e.  B  <->  A  e.  B ) )
32eqcoms 2407 . 2  |-  ( A  =  _V  ->  ( _V  e.  B  <->  A  e.  B ) )
41, 3mtbii 294 1  |-  ( A  =  _V  ->  -.  A  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   _Vcvv 2916
This theorem is referenced by:  afvvdm  27872  afvvfunressn  27874  afvvv  27876  afvvfveq  27879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-v 2918
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