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Theorem nelelne 2432
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 2431 . 2 ((𝐶𝐵 ∧ ¬ 𝐴𝐵) → 𝐶𝐴)
21expcom 115 1 𝐴𝐵 → (𝐶𝐵𝐶𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 2141  wne 2340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-ne 2341
This theorem is referenced by: (None)
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