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Theorem nsstr 38758
Description: If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
nsstr ((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)

Proof of Theorem nsstr
StepHypRef Expression
1 sstr 3591 . . . 4 ((𝐴𝐶𝐶𝐵) → 𝐴𝐵)
21ancoms 469 . . 3 ((𝐶𝐵𝐴𝐶) → 𝐴𝐵)
32adantll 749 . 2 (((¬ 𝐴𝐵𝐶𝐵) ∧ 𝐴𝐶) → 𝐴𝐵)
4 simpll 789 . 2 (((¬ 𝐴𝐵𝐶𝐵) ∧ 𝐴𝐶) → ¬ 𝐴𝐵)
53, 4pm2.65da 599 1 ((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wss 3555
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 3562  df-ss 3569
This theorem is referenced by:  mbfpsssmf  40298
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