Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nvelim Structured version   Unicode version

Theorem nvelim 27968
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4345. (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 4345 . 2  |-  -.  _V  e.  B
2 eleq1 2498 . . 3  |-  ( _V  =  A  ->  ( _V  e.  B  <->  A  e.  B ) )
32eqcoms 2441 . 2  |-  ( A  =  _V  ->  ( _V  e.  B  <->  A  e.  B ) )
41, 3mtbii 295 1  |-  ( A  =  _V  ->  -.  A  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958
This theorem is referenced by:  afvvdm  27995  afvvfunressn  27997  afvvv  27999  afvvfveq  28002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
  Copyright terms: Public domain W3C validator