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Theorem pssnssi 42339
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 4010 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
31, 2mpbi 233 . 2 (𝐴𝐵 ∧ ¬ 𝐵𝐴)
43simpri 489 1 ¬ 𝐵𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wss 3875  wpss 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-v 3417  df-in 3882  df-ss 3892  df-pss 3894
This theorem is referenced by:  nsssmfmbf  44001
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