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Theorem pssnssi 41367
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 4062 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
31, 2mpbi 232 . 2 (𝐴𝐵 ∧ ¬ 𝐵𝐴)
43simpri 488 1 ¬ 𝐵𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 398  wss 3935  wpss 3936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-in 3942  df-ss 3951  df-pss 3953
This theorem is referenced by:  nsssmfmbf  43056
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