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Theorem pssn2lp 3045
 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 3030 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
21simprbi 260 . . 3 (𝐴𝐵 → ¬ 𝐵𝐴)
3 pssss 3039 . . 3 (𝐵𝐴𝐵𝐴)
42, 3nsyl 558 . 2 (𝐴𝐵 → ¬ 𝐵𝐴)
5 imnan 624 . 2 ((𝐴𝐵 → ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
64, 5mpbi 133 1 ¬ (𝐴𝐵𝐵𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ⊆ wss 2917   ⊊ wpss 2918 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933 This theorem is referenced by:  psstr  3049
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