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Theorem cvrnbtwn4 39261
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 32318 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 2735 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 cvrle.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 39253 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌))
5 iman 401 . . . . 5 (((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
6 neanior 3033 . . . . . . . . 9 ((𝑋𝑍𝑍𝑌) ↔ ¬ (𝑋 = 𝑍𝑍 = 𝑌))
76anbi2i 623 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
8 an4 656 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
97, 8bitr3i 277 . . . . . . 7 (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
10 cvrle.l . . . . . . . . . 10 = (le‘𝐾)
1110, 2pltval 18390 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
12113adant3r2 1182 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
1310, 2pltval 18390 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
14133com23 1125 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
15143adant3r1 1181 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
1612, 15anbi12d 632 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌))))
179, 16bitr4id 290 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
1817notbid 318 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
195, 18bitr2id 284 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
20193adant3 1131 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
214, 20mpbid 232 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)))
221, 10posref 18376 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑍𝐵) → 𝑍 𝑍)
23223ad2antr3 1189 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍 𝑍)
24233adant3 1131 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑍 𝑍)
25 breq1 5151 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑍𝑍 𝑍))
2624, 25syl5ibrcom 247 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
271, 10, 3cvrle 39260 . . . . . . . 8 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
2827ex 412 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 𝑌))
29283adant3r3 1183 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 𝑌))
30293impia 1116 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
31 breq2 5152 . . . . 5 (𝑍 = 𝑌 → (𝑋 𝑍𝑋 𝑌))
3230, 31syl5ibrcom 247 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 𝑍))
3326, 32jaod 859 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑋 𝑍))
34 breq1 5151 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑌𝑍 𝑌))
3530, 34syl5ibcom 245 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 𝑌))
36 breq2 5152 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑍𝑍 𝑌))
3724, 36syl5ibcom 245 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3835, 37jaod 859 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑍 𝑌))
3933, 38jcad 512 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → (𝑋 𝑍𝑍 𝑌)))
4021, 39impbid 212 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  ltcplt 18366  ccvr 39244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-proset 18352  df-poset 18371  df-plt 18388  df-covers 39248
This theorem is referenced by:  cvrcmp  39265  leatb  39274  2llnmat  39507  2lnat  39767
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