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Theorem nvelim 27301
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4234. (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 4234 . 2  |-  -.  _V  e.  B
2 eleq1 2418 . . 3  |-  ( _V  =  A  ->  ( _V  e.  B  <->  A  e.  B ) )
32eqcoms 2361 . 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 1642    e. wcel 1710   _Vcvv 2864
This theorem is referenced by:  afvvdm  27329  afvvfunressn  27331  afvvv  27333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-v 2866
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