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Theorem ssneld 3945
 Description: If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
ssneld.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
ssneld (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))

Proof of Theorem ssneld
StepHypRef Expression
1 ssneld.1 . . 3 (𝜑𝐴𝐵)
21sseld 3942 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
32con3d 155 1 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2115   ⊆ wss 3910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-in 3917  df-ss 3927 This theorem is referenced by:  ssneldd  3946  kmlem2  9554  hashbclem  13794  prodss  15280  coprmproddvdslem  15983  mrissmrid  16891  mpfrcl  20274  onsuct0  33797  ftc1anc  35024  dvhdimlem  38626  dvh3dim2  38630  dvh3dim3N  38631  mapdh9a  38971  hdmapval0  39015  hdmap11lem2  39024  iundjiunlem  42921  elbigolo1  44793
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