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Theorem ssnel 39726
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 3747 . 2 ((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
21stoic1a 1845 1 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 2145  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-in 3730  df-ss 3737
This theorem is referenced by:  nelrnres  39893  supminfxr2  40212
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