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Theorem elnelne2 3051
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 3041 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 3034 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 587 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 2155  wne 2937  wnel 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-cleq 2758  df-clel 2761  df-ne 2938  df-nel 3041
This theorem is referenced by:  nelrnfvne  6543  eldmrexrnb  6556  absprodnn  15612  frgrncvvdeqlem2  27580  frgrncvvdeqlem3  27581  afv0nbfvbi  41899  2zrngnmlid  42618  2zrngnmrid  42619
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