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Theorem plttr 18060
Description: The less-than relation is transitive. (psstr 4039 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltnlt.b 𝐵 = (Base‘𝐾)
pltnlt.s < = (lt‘𝐾)
Assertion
Ref Expression
plttr ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Proof of Theorem plttr
StepHypRef Expression
1 eqid 2738 . . . . . 6 (le‘𝐾) = (le‘𝐾)
2 pltnlt.s . . . . . 6 < = (lt‘𝐾)
31, 2pltle 18051 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋(le‘𝐾)𝑌))
433adant3r3 1183 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌𝑋(le‘𝐾)𝑌))
51, 2pltle 18051 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 < 𝑍𝑌(le‘𝐾)𝑍))
653adant3r1 1181 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 < 𝑍𝑌(le‘𝐾)𝑍))
7 pltnlt.b . . . . 5 𝐵 = (Base‘𝐾)
87, 1postr 18038 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑍) → 𝑋(le‘𝐾)𝑍))
94, 6, 8syl2and 608 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋(le‘𝐾)𝑍))
107, 2pltn2lp 18059 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
11103adant3r3 1183 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
12 breq2 5078 . . . . . . 7 (𝑋 = 𝑍 → (𝑌 < 𝑋𝑌 < 𝑍))
1312anbi2d 629 . . . . . 6 (𝑋 = 𝑍 → ((𝑋 < 𝑌𝑌 < 𝑋) ↔ (𝑋 < 𝑌𝑌 < 𝑍)))
1413notbid 318 . . . . 5 (𝑋 = 𝑍 → (¬ (𝑋 < 𝑌𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1511, 14syl5ibcom 244 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍 → ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1615necon2ad 2958 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋𝑍))
179, 16jcad 513 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → (𝑋(le‘𝐾)𝑍𝑋𝑍)))
181, 2pltval 18050 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍𝑋𝑍)))
19183adant3r2 1182 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍𝑋𝑍)))
2017, 19sylibrd 258 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943   class class class wbr 5074  cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  ltcplt 18026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-proset 18013  df-poset 18031  df-plt 18048
This theorem is referenced by:  pltletr  18061  plelttr  18062  pospo  18063  archiabllem2c  31449  ofldchr  31513  hlhgt2  37403  hl0lt1N  37404  lhp0lt  38017
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