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Theorem ssneldd 3131
Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssneld.1 (𝜑𝐴𝐵)
ssneldd.2 (𝜑 → ¬ 𝐶𝐵)
Assertion
Ref Expression
ssneldd (𝜑 → ¬ 𝐶𝐴)

Proof of Theorem ssneldd
StepHypRef Expression
1 ssneldd.2 . 2 (𝜑 → ¬ 𝐶𝐵)
2 ssneld.1 . . 3 (𝜑𝐴𝐵)
32ssneld 3130 . 2 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
41, 3mpd 13 1 (𝜑 → ¬ 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:  0nelrel  4632  addnqprlemfl  7479  addnqprlemfu  7480  mulnqprlemfl  7495  mulnqprlemfu  7496  cauappcvgprlemladdru  7576  fprodntrivap  11481  fprodssdc  11487
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