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

Theorem neneor 3118
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 2843 . . 3 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 3041 . 2 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
3 neorian 3111 . 2 ((𝐴𝐶𝐵𝐶) ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶))
42, 3sylibr 235 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wne 3016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-cleq 2814  df-ne 3017
This theorem is referenced by:  wemapso2lem  9005  trgcopyeulem  26519
  Copyright terms: Public domain W3C validator