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

Theorem nvelim 47135
Description: If a class is the universal class it doesn't belong to any class, generalization of nvel 5316. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nvelim (𝐴 = V → ¬ 𝐴𝐵)

Proof of Theorem nvelim
StepHypRef Expression
1 nvel 5316 . 2 ¬ V ∈ 𝐵
2 eleq1 2829 . . 3 (V = 𝐴 → (V ∈ 𝐵𝐴𝐵))
32eqcoms 2745 . 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 1540  wcel 2108  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482
This theorem is referenced by:  afvvdm  47153  afvvfunressn  47155  afvvv  47157  afvvfveq  47160
  Copyright terms: Public domain W3C validator