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Theorem neneor 3098
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 2848 . . 3 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 3024 . 2 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
3 neorian 3093 . 2 ((𝐴𝐶𝐵𝐶) ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶))
42, 3sylibr 226 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 880   = wceq 1658  wne 2999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-cleq 2818  df-ne 3000
This theorem is referenced by:  trgcopyeulem  26114
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