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Theorem pssn2lp 4062
Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
pssn2lp ¬ (𝐴𝐵𝐵𝐴)

Proof of Theorem pssn2lp
StepHypRef Expression
1 dfpss3 4047 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
21simprbi 498 . . 3 (𝐴𝐵 → ¬ 𝐵𝐴)
3 pssss 4056 . . 3 (𝐵𝐴𝐵𝐴)
42, 3nsyl 140 . 2 (𝐴𝐵 → ¬ 𝐵𝐴)
5 imnan 401 . 2 ((𝐴𝐵 → ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
64, 5mpbi 229 1 ¬ (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wss 3911  wpss 3912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3446  df-in 3918  df-ss 3928  df-pss 3930
This theorem is referenced by:  psstr  4065  cvnsym  31274
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