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Theorem nvelim 47030
Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5318. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nvelim (𝐴 = V → ¬ 𝐴𝐵)

Proof of Theorem nvelim
StepHypRef Expression
1 nvel 5318 . 2 ¬ V ∈ 𝐵
2 eleq1 2825 . . 3 (V = 𝐴 → (V ∈ 𝐵𝐴𝐵))
32eqcoms 2741 . 2 (𝐴 = V → (V ∈ 𝐵𝐴𝐵))
41, 3mtbii 326 1 (𝐴 = V → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1535  wcel 2104  Vcvv 3477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-sep 5301
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479
This theorem is referenced by:  afvvdm  47048  afvvfunressn  47050  afvvv  47052  afvvfveq  47055
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