MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neneor Structured version   Visualization version   GIF version

Theorem neneor 2880
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 2630 . . 3 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 2806 . 2 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
3 neorian 2875 . 2 ((𝐴𝐶𝐵𝐶) ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶))
42, 3sylibr 222 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382   = wceq 1474  wne 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-cleq 2602  df-ne 2781
This theorem is referenced by:  trgcopyeulem  25442
  Copyright terms: Public domain W3C validator