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Theorem nelelne 3035
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 3034 . 2 ((𝐶𝐵 ∧ ¬ 𝐴𝐵) → 𝐶𝐴)
21expcom 413 1 𝐴𝐵 → (𝐶𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2098  wne 2934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-clel 2804  df-ne 2935
This theorem is referenced by:  difsn  4796  elneq  9592  frgrncvvdeqlem7  30063  frgrncvvdeqlem9  30065  prter2  38262
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