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Theorem ssneldd 2921
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 2920 . 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 1370  wss 2890
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 532  ax-in2 533  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904
This theorem is referenced by: (None)
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