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Theorem plelttr 17653
 Description: Transitive law for chained "less than or equal to" and "less than". (sspsstr 4013 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 17646 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
543adant3r3 1181 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
61, 3plttr 17651 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
76expd 419 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
8 breq1 5038 . . . . . 6 (𝑋 = 𝑌 → (𝑋 < 𝑍𝑌 < 𝑍))
98biimprd 251 . . . . 5 (𝑋 = 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍))
109a1i 11 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
117, 10jaod 856 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑋 = 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍)))
125, 11sylbid 243 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
1312impd 414 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5035  ‘cfv 6339  Basecbs 16546  lecple 16635  Posetcpo 17621  ltcplt 17622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347  df-proset 17609  df-poset 17627  df-plt 17639 This theorem is referenced by:  isarchi3  30971  archiabllem2c  30979  athgt  37058  1cvratex  37075
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