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Theorem neneor 3031
 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 2781 . . 3 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
21necon3ai 2957 . 2 (𝐴𝐵 → ¬ (𝐴 = 𝐶𝐵 = 𝐶))
3 neorian 3026 . 2 ((𝐴𝐶𝐵𝐶) ↔ ¬ (𝐴 = 𝐶𝐵 = 𝐶))
42, 3sylibr 224 1 (𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1632   ≠ wne 2932 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-cleq 2753  df-ne 2933 This theorem is referenced by:  trgcopyeulem  25917
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