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Theorem ssneldd 3070
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 3069 . 2 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
41, 3mpd 13 1 (𝜑 → ¬ 𝐶𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1465  wss 3041
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 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by:  0nelrel  4555  addnqprlemfl  7335  addnqprlemfu  7336  mulnqprlemfl  7351  mulnqprlemfu  7352  cauappcvgprlemladdru  7432
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