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Theorem ssneldd 3276
 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 (φA B)
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 (φA B)
32ssneld 3275 . 2 (φ → (¬ C B → ¬ C A))
41, 3mpd 14 1 (φ → ¬ C A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1710   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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