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Theorem nvelim 42726
Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5077. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nvelim (𝐴 = V → ¬ 𝐴𝐵)

Proof of Theorem nvelim
StepHypRef Expression
1 nvel 5077 . 2 ¬ V ∈ 𝐵
2 eleq1 2854 . . 3 (V = 𝐴 → (V ∈ 𝐵𝐴𝐵))
32eqcoms 2787 . 2 (𝐴 = V → (V ∈ 𝐵𝐴𝐵))
41, 3mtbii 318 1 (𝐴 = V → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198   = wceq 1507  wcel 2050  Vcvv 3416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2751  ax-sep 5060
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-v 3418
This theorem is referenced by:  afvvdm  42744  afvvfunressn  42746  afvvv  42748  afvvfveq  42751
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