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Theorem ssneld 3130
 Description: If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
ssneld.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
ssneld (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))

Proof of Theorem ssneld
StepHypRef Expression
1 ssneld.1 . . 3 (𝜑𝐴𝐵)
21sseld 3127 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
32con3d 621 1 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2128   ⊆ wss 3102 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 604  ax-in2 605  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115 This theorem is referenced by:  ssneldd  3131  sumdc  11237  summodclem2a  11260  zsumdc  11263  isumss2  11272  zproddc  11458  prodssdc  11468  decidin  13330
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