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Theorem pssnssi 44498
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 4086 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
31, 2mpbi 229 . 2 (𝐴𝐵 ∧ ¬ 𝐵𝐴)
43simpri 484 1 ¬ 𝐵𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394  wss 3949  wpss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3475  df-in 3956  df-ss 3966  df-pss 3968
This theorem is referenced by:  nsssmfmbf  46196
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