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Theorem ssneld 3159
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 3156 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
32con3d 631 1 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 2148  wss 3131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144
This theorem is referenced by:  ssneldd  3160  sumdc  11368  summodclem2a  11391  zsumdc  11394  isumss2  11403  zproddc  11589  prodssdc  11599  decidin  14588
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