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Theorem plttr 18387
Description: The less-than relation is transitive. (psstr 4107 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 2737 . . . . . 6 (le‘𝐾) = (le‘𝐾)
2 pltnlt.s . . . . . 6 < = (lt‘𝐾)
31, 2pltle 18378 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋(le‘𝐾)𝑌))
433adant3r3 1185 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌𝑋(le‘𝐾)𝑌))
51, 2pltle 18378 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 < 𝑍𝑌(le‘𝐾)𝑍))
653adant3r1 1183 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 < 𝑍𝑌(le‘𝐾)𝑍))
7 pltnlt.b . . . . 5 𝐵 = (Base‘𝐾)
87, 1postr 18366 . . . 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 18386 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
11103adant3r3 1185 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
12 breq2 5147 . . . . . . 7 (𝑋 = 𝑍 → (𝑌 < 𝑋𝑌 < 𝑍))
1312anbi2d 630 . . . . . 6 (𝑋 = 𝑍 → ((𝑋 < 𝑌𝑌 < 𝑋) ↔ (𝑋 < 𝑌𝑌 < 𝑍)))
1413notbid 318 . . . . 5 (𝑋 = 𝑍 → (¬ (𝑋 < 𝑌𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1511, 14syl5ibcom 245 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍 → ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1615necon2ad 2955 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋𝑍))
179, 16jcad 512 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → (𝑋(le‘𝐾)𝑍𝑋𝑍)))
181, 2pltval 18377 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍𝑋𝑍)))
19183adant3r2 1184 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍𝑋𝑍)))
2017, 19sylibrd 259 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  ltcplt 18354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-proset 18340  df-poset 18359  df-plt 18375
This theorem is referenced by:  pltletr  18388  plelttr  18389  pospo  18390  archiabllem2c  33202  ofldchr  33344  hlhgt2  39391  hl0lt1N  39392  lhp0lt  40005
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