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

Theorem pssnssi 41244
Description: A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
pssnssi.1 𝐴𝐵
Assertion
Ref Expression
pssnssi ¬ 𝐵𝐴

Proof of Theorem pssnssi
StepHypRef Expression
1 pssnssi.1 . . 3 𝐴𝐵
2 dfpss3 4060 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
31, 2mpbi 231 . 2 (𝐴𝐵 ∧ ¬ 𝐵𝐴)
43simpri 486 1 ¬ 𝐵𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wss 3933  wpss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-in 3940  df-ss 3949  df-pss 3951
This theorem is referenced by:  nsssmfmbf  42932
  Copyright terms: Public domain W3C validator