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Theorem nvelim 28080
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4169. (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 4169 . 2  |-  -.  _V  e.  B
2 eleq1 2356 . . 3  |-  ( _V  =  A  ->  ( _V  e.  B  <->  A  e.  B ) )
32eqcoms 2299 . 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 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  afvvdm  28108  afvvfunressn  28110  afvvv  28112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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