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Theorem cvrnbtwn4 34046
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 28997 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 2621 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 cvrle.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 34038 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌))
5 iman 440 . . . . 5 (((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
6 cvrle.l . . . . . . . . . 10 = (le‘𝐾)
76, 2pltval 16881 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3r2 1272 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
96, 2pltval 16881 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
1093com23 1268 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
11103adant3r1 1271 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
128, 11anbi12d 746 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌))))
13 neanior 2882 . . . . . . . . 9 ((𝑋𝑍𝑍𝑌) ↔ ¬ (𝑋 = 𝑍𝑍 = 𝑌))
1413anbi2i 729 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
15 an4 864 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1614, 15bitr3i 266 . . . . . . 7 (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1712, 16syl6rbbr 279 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
1817notbid 308 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
195, 18syl5rbb 273 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
20193adant3 1079 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
214, 20mpbid 222 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)))
221, 6posref 16872 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑍𝐵) → 𝑍 𝑍)
23223ad2antr3 1226 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍 𝑍)
24233adant3 1079 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑍 𝑍)
25 breq1 4616 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑍𝑍 𝑍))
2624, 25syl5ibrcom 237 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
271, 6, 3cvrle 34045 . . . . . . . 8 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
2827ex 450 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 𝑌))
29283adant3r3 1273 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 𝑌))
30293impia 1258 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
31 breq2 4617 . . . . 5 (𝑍 = 𝑌 → (𝑋 𝑍𝑋 𝑌))
3230, 31syl5ibrcom 237 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 𝑍))
3326, 32jaod 395 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑋 𝑍))
34 breq1 4616 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑌𝑍 𝑌))
3530, 34syl5ibcom 235 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 𝑌))
36 breq2 4617 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑍𝑍 𝑌))
3724, 36syl5ibcom 235 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3835, 37jaod 395 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑍 𝑌))
3933, 38jcad 555 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → (𝑋 𝑍𝑍 𝑌)))
4021, 39impbid 202 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  Posetcpo 16861  ltcplt 16862  ccvr 34029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-preset 16849  df-poset 16867  df-plt 16879  df-covers 34033
This theorem is referenced by:  cvrcmp  34050  leatb  34059  2llnmat  34290  2lnat  34550
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