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Theorem nelelne 3040
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 3039 . 2 ((𝐶𝐵 ∧ ¬ 𝐴𝐵) → 𝐶𝐴)
21expcom 414 1 𝐴𝐵 → (𝐶𝐵𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2105  wne 2940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-cleq 2728  df-clel 2814  df-ne 2941
This theorem is referenced by:  difsn  4745  elneq  9455  frgrncvvdeqlem7  28957  frgrncvvdeqlem9  28959  prter2  37148
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