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Theorem ssneldd 2942
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 → ¬ 𝐶 B)
Assertion
Ref Expression
ssneldd (φ → ¬ 𝐶 A)

Proof of Theorem ssneldd
StepHypRef Expression
1 ssneldd.2 . 2 (φ → ¬ 𝐶 B)
2 ssneld.1 . . 3 (φAB)
32ssneld 2941 . 2 (φ → (¬ 𝐶 B → ¬ 𝐶 A))
41, 3mpd 13 1 (φ → ¬ 𝐶 A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wcel 1390  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925
This theorem is referenced by:  addnqpr1lemil  6539  addnqpr1lemiu  6540
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