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Theorem pltn2lp 18336
Description: The less-than relation has no 2-cycle loops. (pssn2lp 4097 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 2725 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 pltnlt.s . . . . 5 < = (lt‘𝐾)
41, 2, 3pltnle 18333 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋)
54ex 411 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌(le‘𝐾)𝑋))
62, 3pltle 18328 . . . 4 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑋𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
763com23 1123 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
85, 7nsyld 156 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 < 𝑋))
9 imnan 398 . 2 ((𝑋 < 𝑌 → ¬ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌𝑌 < 𝑋))
108, 9sylib 217 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243  Posetcpo 18302  ltcplt 18303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-proset 18290  df-poset 18308  df-plt 18325
This theorem is referenced by:  plttr  18337
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