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Theorem nvelim 38334
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4560. (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 4560 . 2  |-  -.  _V  e.  B
2 eleq1 2494 . . 3  |-  ( _V  =  A  ->  ( _V  e.  B  <->  A  e.  B ) )
32eqcoms 2434 . 2  |-  ( A  =  _V  ->  ( _V  e.  B  <->  A  e.  B ) )
41, 3mtbii 303 1  |-  ( A  =  _V  ->  -.  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1868   _Vcvv 3081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3083
This theorem is referenced by:  afvvdm  38355  afvvfunressn  38357  afvvv  38359  afvvfveq  38362
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