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Theorem nelelne 2875
Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)
Assertion
Ref Expression
nelelne 𝐴𝐵 → (𝐶𝐵𝐶𝐴))

Proof of Theorem nelelne
StepHypRef Expression
1 nelne2 2874 . 2 ((𝐶𝐵 ∧ ¬ 𝐴𝐵) → 𝐶𝐴)
21expcom 449 1 𝐴𝐵 → (𝐶𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1975  wne 2775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-cleq 2598  df-clel 2601  df-ne 2777
This theorem is referenced by:  difsn  4264  prter2  32983  frgrncvvdeqlemA  41474  frgrncvvdeqlemC  41476
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