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Theorem ssnel 38227
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 3563 . 2 ((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
21stoic1a 1688 1 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 1977  wss 3540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554
This theorem is referenced by:  nelrnres  38369
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