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Theorem pltn2lp 17847
Description: The less-than relation has no 2-cycle loops. (pssn2lp 4016 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 2737 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 pltnlt.s . . . . 5 < = (lt‘𝐾)
41, 2, 3pltnle 17844 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋)
54ex 416 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌(le‘𝐾)𝑋))
62, 3pltle 17839 . . . 4 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑋𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
763com23 1128 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
85, 7nsyld 159 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 < 𝑋))
9 imnan 403 . 2 ((𝑋 < 𝑌 → ¬ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌𝑌 < 𝑋))
108, 9sylib 221 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110   class class class wbr 5053  cfv 6380  Basecbs 16760  lecple 16809  Posetcpo 17814  ltcplt 17815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-proset 17802  df-poset 17820  df-plt 17836
This theorem is referenced by:  plttr  17848
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