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Theorem pssn2lp 3073
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 3058 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
21simprbi 264 . . 3 (𝐴𝐵 → ¬ 𝐵𝐴)
3 pssss 3067 . . 3 (𝐵𝐴𝐵𝐴)
42, 3nsyl 568 . 2 (𝐴𝐵 → ¬ 𝐵𝐴)
5 imnan 634 . 2 ((𝐴𝐵 → ¬ 𝐵𝐴) ↔ ¬ (𝐴𝐵𝐵𝐴))
64, 5mpbi 137 1 ¬ (𝐴𝐵𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wss 2945  wpss 2946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ne 2221  df-in 2952  df-ss 2959  df-pss 2961
This theorem is referenced by:  psstr  3077
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