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Theorem plelttr 18389
Description: Transitive law for chained "less than or equal to" and "less than". (sspsstr 4108 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 18382 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
543adant3r3 1185 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
61, 3plttr 18387 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
76expd 415 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
8 breq1 5146 . . . . . 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 860 . . 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 848  w3a 1087   = wceq 1540  wcel 2108   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:  isarchi3  33194  archiabllem2c  33202  athgt  39458  1cvratex  39475
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