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Theorem ssnel 38722
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 3582 . 2 ((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
21stoic1a 1694 1 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1987  wss 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3566  df-ss 3573
This theorem is referenced by:  nelrnres  38879
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