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Theorem plelttr 18402
Description: Transitive law for chained "less than or equal to" and "less than". (sspsstr 4118 analog.) (Contributed by NM, 2-May-2012.)
Hypotheses
Ref Expression
pltletr.b 𝐵 = (Base‘𝐾)
pltletr.l = (le‘𝐾)
pltletr.s < = (lt‘𝐾)
Assertion
Ref Expression
plelttr ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Proof of Theorem plelttr
StepHypRef Expression
1 pltletr.b . . . . 5 𝐵 = (Base‘𝐾)
2 pltletr.l . . . . 5 = (le‘𝐾)
3 pltletr.s . . . . 5 < = (lt‘𝐾)
41, 2, 3pleval2 18395 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
543adant3r3 1183 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
61, 3plttr 18400 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
76expd 415 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
8 breq1 5151 . . . . . 6 (𝑋 = 𝑌 → (𝑋 < 𝑍𝑌 < 𝑍))
98biimprd 248 . . . . 5 (𝑋 = 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍))
109a1i 11 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
117, 10jaod 859 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑋 = 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍)))
125, 11sylbid 240 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
1312impd 410 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  ltcplt 18366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-proset 18352  df-poset 18371  df-plt 18388
This theorem is referenced by:  isarchi3  33177  archiabllem2c  33185  athgt  39439  1cvratex  39456
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