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Theorem pltn2lp 18387
Description: The less-than relation has no 2-cycle loops. (pssn2lp 4103 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltnlt.b 𝐵 = (Base‘𝐾)
pltnlt.s < = (lt‘𝐾)
Assertion
Ref Expression
pltn2lp ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))

Proof of Theorem pltn2lp
StepHypRef Expression
1 pltnlt.b . . . . 5 𝐵 = (Base‘𝐾)
2 eqid 2736 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 pltnlt.s . . . . 5 < = (lt‘𝐾)
41, 2, 3pltnle 18384 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋)
54ex 412 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌(le‘𝐾)𝑋))
62, 3pltle 18379 . . . 4 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑋𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
763com23 1126 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
85, 7nsyld 156 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 < 𝑋))
9 imnan 399 . 2 ((𝑋 < 𝑌 → ¬ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌𝑌 < 𝑋))
108, 9sylib 218 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  Basecbs 17248  lecple 17305  Posetcpo 18354  ltcplt 18355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-proset 18341  df-poset 18360  df-plt 18376
This theorem is referenced by:  plttr  18388
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