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Theorem neneor 3042
Description: If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)
Assertion
Ref Expression
neneor (𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Proof of Theorem neneor
StepHypRef Expression
1 eqtr3 2762 . . 3 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 2966 . 2 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
3 neorian 3037 . 2 ((𝐴𝐶𝐵𝐶) ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶))
42, 3sylibr 233 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wne 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-cleq 2728  df-ne 2942
This theorem is referenced by:  wemapso2lem  9484  trgcopyeulem  27633
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