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Theorem elnelne2 2894
 Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
elnelne2 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)

Proof of Theorem elnelne2
StepHypRef Expression
1 df-nel 2783 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 2879 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 491 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-ne 2782  df-nel 2783 This theorem is referenced by:  nelrnfvne  6261  eldmrexrnb  6274  absprodnn  15169  afv0nbfvbi  39880  2zrngnmlid  41739  2zrngnmrid  41740
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