Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nsstr Structured version   Visualization version   GIF version

Theorem nsstr 41512
 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 3950 . . . 4 ((𝐴𝐶𝐶𝐵) → 𝐴𝐵)
21ancoms 461 . . 3 ((𝐶𝐵𝐴𝐶) → 𝐴𝐵)
32adantll 712 . 2 (((¬ 𝐴𝐵𝐶𝐵) ∧ 𝐴𝐶) → 𝐴𝐵)
4 simpll 765 . 2 (((¬ 𝐴𝐵𝐶𝐵) ∧ 𝐴𝐶) → ¬ 𝐴𝐵)
53, 4pm2.65da 815 1 ((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ⊆ wss 3909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3472  df-in 3916  df-ss 3926 This theorem is referenced by:  mbfpsssmf  43203
 Copyright terms: Public domain W3C validator