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Theorem pssn2lp 3664
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 3649 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
21simprbi 478 . . 3 (𝐴𝐵 → ¬ 𝐵𝐴)
3 pssss 3658 . . 3 (𝐵𝐴𝐵𝐴)
42, 3nsyl 133 . 2 (𝐴𝐵 → ¬ 𝐵𝐴)
5 imnan 436 . 2 ((𝐴𝐵 → ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
64, 5mpbi 218 1 ¬ (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wss 3534  wpss 3535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-ne 2776  df-in 3541  df-ss 3548  df-pss 3550
This theorem is referenced by:  psstr  3667  cvnsym  28334
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