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Theorem cvrnbtwn4 35433
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 29720 analog.) (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrle.b 𝐵 = (Base‘𝐾)
cvrle.l = (le‘𝐾)
cvrle.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrnbtwn4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))

Proof of Theorem cvrnbtwn4
StepHypRef Expression
1 cvrle.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2778 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 cvrle.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 35425 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌))
5 iman 392 . . . . 5 (((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
6 cvrle.l . . . . . . . . . 10 = (le‘𝐾)
76, 2pltval 17346 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3r2 1191 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
96, 2pltval 17346 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
1093com23 1117 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
11103adant3r1 1190 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
128, 11anbi12d 624 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌))))
13 neanior 3062 . . . . . . . . 9 ((𝑋𝑍𝑍𝑌) ↔ ¬ (𝑋 = 𝑍𝑍 = 𝑌))
1413anbi2i 616 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
15 an4 646 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1614, 15bitr3i 269 . . . . . . 7 (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1712, 16syl6rbbr 282 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
1817notbid 310 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
195, 18syl5rbb 276 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
20193adant3 1123 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
214, 20mpbid 224 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)))
221, 6posref 17337 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑍𝐵) → 𝑍 𝑍)
23223ad2antr3 1198 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍 𝑍)
24233adant3 1123 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑍 𝑍)
25 breq1 4889 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑍𝑍 𝑍))
2624, 25syl5ibrcom 239 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
271, 6, 3cvrle 35432 . . . . . . . 8 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
2827ex 403 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 𝑌))
29283adant3r3 1192 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 𝑌))
30293impia 1106 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
31 breq2 4890 . . . . 5 (𝑍 = 𝑌 → (𝑋 𝑍𝑋 𝑌))
3230, 31syl5ibrcom 239 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 𝑍))
3326, 32jaod 848 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑋 𝑍))
34 breq1 4889 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑌𝑍 𝑌))
3530, 34syl5ibcom 237 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 𝑌))
36 breq2 4890 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑍𝑍 𝑌))
3724, 36syl5ibcom 237 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3835, 37jaod 848 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑍 𝑌))
3933, 38jcad 508 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → (𝑋 𝑍𝑍 𝑌)))
4021, 39impbid 204 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836  w3a 1071   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4886  cfv 6135  Basecbs 16255  lecple 16345  Posetcpo 17326  ltcplt 17327  ccvr 35416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-proset 17314  df-poset 17332  df-plt 17344  df-covers 35420
This theorem is referenced by:  cvrcmp  35437  leatb  35446  2llnmat  35678  2lnat  35938
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