Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pssn2lp Structured version   Visualization version   GIF version

Theorem pssn2lp 3670
 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 3655 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
21simprbi 479 . . 3 (𝐴𝐵 → ¬ 𝐵𝐴)
3 pssss 3664 . . 3 (𝐵𝐴𝐵𝐴)
42, 3nsyl 134 . 2 (𝐴𝐵 → ¬ 𝐵𝐴)
5 imnan 437 . 2 ((𝐴𝐵 → ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
64, 5mpbi 219 1 ¬ (𝐴𝐵𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by:  psstr  3673  cvnsym  28533
 Copyright terms: Public domain W3C validator