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Theorem nsstr 39791
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 3753 . . . 4 ((𝐴𝐶𝐶𝐵) → 𝐴𝐵)
21ancoms 468 . . 3 ((𝐶𝐵𝐴𝐶) → 𝐴𝐵)
32adantll 752 . 2 (((¬ 𝐴𝐵𝐶𝐵) ∧ 𝐴𝐶) → 𝐴𝐵)
4 simpll 807 . 2 (((¬ 𝐴𝐵𝐶𝐵) ∧ 𝐴𝐶) → ¬ 𝐴𝐵)
53, 4pm2.65da 601 1 ((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-in 3723  df-ss 3730
This theorem is referenced by:  mbfpsssmf  41516
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