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Theorem ssnel 44980
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 3989 . 2 ((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
21stoic1a 1768 1 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wss 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-clel 2813  df-ss 3979
This theorem is referenced by:  nelrnres  45129  supminfxr2  45418
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