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Theorem nelelne 3030
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 3029 . 2 ((𝐶𝐵 ∧ ¬ 𝐴𝐵) → 𝐶𝐴)
21expcom 450 1 𝐴𝐵 → (𝐶𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2139  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-an 385  df-ex 1854  df-cleq 2753  df-clel 2756  df-ne 2933
This theorem is referenced by:  ssdifsnOLD  4464  difsn  4473  elneq  8668  frgrncvvdeqlem7  27459  frgrncvvdeqlem9  27461  prter2  34670
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