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Theorem ssnel 39964
Description: If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
ssnel ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)

Proof of Theorem ssnel
StepHypRef Expression
1 ssel2 3793 . 2 ((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
21stoic1a 1868 1 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wcel 2157  wss 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-in 3776  df-ss 3783
This theorem is referenced by:  nelrnres  40128  supminfxr2  40442
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