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Theorem elnelne2 3057
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 3047 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 3039 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 595 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  wne 2940  wnel 3046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-clel 2811  df-ne 2941  df-nel 3047
This theorem is referenced by:  nelrnfvne  7029  eldmrexrnb  7043  absprodnn  16499  frgrncvvdeqlem2  29286  frgrncvvdeqlem3  29287  afv0nbfvbi  45469  uniimaelsetpreimafv  45674  imasetpreimafvbijlemfv1  45681  2zrngnmlid  46333  2zrngnmrid  46334
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