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Theorem nvelim 41716
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4931. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nvelim (𝐴 = V → ¬ 𝐴𝐵)

Proof of Theorem nvelim
StepHypRef Expression
1 nvel 4931 . 2 ¬ V ∈ 𝐵
2 eleq1 2838 . . 3 (V = 𝐴 → (V ∈ 𝐵𝐴𝐵))
32eqcoms 2779 . 2 (𝐴 = V → (V ∈ 𝐵𝐴𝐵))
41, 3mtbii 315 1 (𝐴 = V → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  afvvdm  41737  afvvfunressn  41739  afvvv  41741  afvvfveq  41744
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