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Theorem pssn2lp 3370
 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 ¬ (AB BA)

Proof of Theorem pssn2lp
StepHypRef Expression
1 dfpss3 3355 . . . 4 (AB ↔ (A B ¬ B A))
21simprbi 450 . . 3 (AB → ¬ B A)
3 pssss 3364 . . 3 (BAB A)
42, 3nsyl 113 . 2 (AB → ¬ BA)
5 imnan 411 . 2 ((AB → ¬ BA) ↔ ¬ (AB BA))
64, 5mpbi 199 1 ¬ (AB BA)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  psstr  3373
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