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Theorem cvrnbtwn4 35167
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 29539 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 2765 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 cvrle.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 35159 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌))
5 iman 390 . . . . 5 (((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
6 cvrle.l . . . . . . . . . 10 = (le‘𝐾)
76, 2pltval 17228 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3r2 1234 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
96, 2pltval 17228 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
1093com23 1156 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
11103adant3r1 1233 . . . . . . . 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 3029 . . . . . . . . 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 268 . . . . . . 7 (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1712, 16syl6rbbr 281 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
1817notbid 309 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
195, 18syl5rbb 275 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
20193adant3 1162 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
214, 20mpbid 223 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)))
221, 6posref 17219 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑍𝐵) → 𝑍 𝑍)
23223ad2antr3 1241 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍 𝑍)
24233adant3 1162 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑍 𝑍)
25 breq1 4812 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑍𝑍 𝑍))
2624, 25syl5ibrcom 238 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
271, 6, 3cvrle 35166 . . . . . . . 8 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
2827ex 401 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 𝑌))
29283adant3r3 1235 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 𝑌))
30293impia 1145 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
31 breq2 4813 . . . . 5 (𝑍 = 𝑌 → (𝑋 𝑍𝑋 𝑌))
3230, 31syl5ibrcom 238 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 𝑍))
3326, 32jaod 885 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑋 𝑍))
34 breq1 4812 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑌𝑍 𝑌))
3530, 34syl5ibcom 236 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 𝑌))
36 breq2 4813 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑍𝑍 𝑌))
3724, 36syl5ibcom 236 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3835, 37jaod 885 . . 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 203 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  Basecbs 16132  lecple 16223  Posetcpo 17208  ltcplt 17209  ccvr 35150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-proset 17196  df-poset 17214  df-plt 17226  df-covers 35154
This theorem is referenced by:  cvrcmp  35171  leatb  35180  2llnmat  35412  2lnat  35672
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