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Theorem nvelim 39756
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4624. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nvelim (𝐴 = V → ¬ 𝐴𝐵)

Proof of Theorem nvelim
StepHypRef Expression
1 nvel 4624 . 2 ¬ V ∈ 𝐵
2 eleq1 2580 . . 3 (V = 𝐴 → (V ∈ 𝐵𝐴𝐵))
32eqcoms 2522 . 2 (𝐴 = V → (V ∈ 𝐵𝐴𝐵))
41, 3mtbii 314 1 (𝐴 = V → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194   = wceq 1474  wcel 1938  Vcvv 3077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-v 3079
This theorem is referenced by:  afvvdm  39778  afvvfunressn  39780  afvvv  39782  afvvfveq  39785
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