Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrnbtwn4 Structured version   Visualization version   GIF version

Theorem cvrnbtwn4 36297
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 29994 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 2821 . . . 4 (lt‘𝐾) = (lt‘𝐾)
3 cvrle.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 36289 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌))
5 iman 402 . . . . 5 (((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
6 cvrle.l . . . . . . . . . 10 = (le‘𝐾)
76, 2pltval 17560 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3r2 1175 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
96, 2pltval 17560 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
1093com23 1118 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
11103adant3r1 1174 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
128, 11anbi12d 630 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌))))
13 neanior 3109 . . . . . . . . 9 ((𝑋𝑍𝑍𝑌) ↔ ¬ (𝑋 = 𝑍𝑍 = 𝑌))
1413anbi2i 622 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)))
15 an4 652 . . . . . . . 8 (((𝑋 𝑍𝑍 𝑌) ∧ (𝑋𝑍𝑍𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1614, 15bitr3i 278 . . . . . . 7 (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ((𝑋 𝑍𝑋𝑍) ∧ (𝑍 𝑌𝑍𝑌)))
1712, 16syl6rbbr 291 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
1817notbid 319 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 𝑍𝑍 𝑌) ∧ ¬ (𝑋 = 𝑍𝑍 = 𝑌)) ↔ ¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌)))
195, 18syl5rbb 285 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
20193adant3 1124 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋(lt‘𝐾)𝑍𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌))))
214, 20mpbid 233 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) → (𝑋 = 𝑍𝑍 = 𝑌)))
221, 6posref 17551 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑍𝐵) → 𝑍 𝑍)
23223ad2antr3 1182 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍 𝑍)
24233adant3 1124 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑍 𝑍)
25 breq1 5061 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑍𝑍 𝑍))
2624, 25syl5ibrcom 248 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
271, 6, 3cvrle 36296 . . . . . . . 8 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
2827ex 413 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 𝑌))
29283adant3r3 1176 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 𝑌))
30293impia 1109 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
31 breq2 5062 . . . . 5 (𝑍 = 𝑌 → (𝑋 𝑍𝑋 𝑌))
3230, 31syl5ibrcom 248 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 𝑍))
3326, 32jaod 853 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑋 𝑍))
34 breq1 5061 . . . . 5 (𝑋 = 𝑍 → (𝑋 𝑌𝑍 𝑌))
3530, 34syl5ibcom 246 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 𝑌))
36 breq2 5062 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑍𝑍 𝑌))
3724, 36syl5ibcom 246 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3835, 37jaod 853 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → 𝑍 𝑌))
3933, 38jcad 513 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍𝑍 = 𝑌) → (𝑋 𝑍𝑍 𝑌)))
4021, 39impbid 213 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3016   class class class wbr 5058  cfv 6349  Basecbs 16473  lecple 16562  Posetcpo 17540  ltcplt 17541  ccvr 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-proset 17528  df-poset 17546  df-plt 17558  df-covers 36284
This theorem is referenced by:  cvrcmp  36301  leatb  36310  2llnmat  36542  2lnat  36802
  Copyright terms: Public domain W3C validator