Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrnbtwn2 Structured version   Visualization version   GIF version

Theorem cvrnbtwn2 35231
Description: The covers relation implies no in-betweenness. (cvnbtwn2 29602 analog.) (Contributed by NM, 17-Nov-2011.)
Hypotheses
Ref Expression
cvrletr.b 𝐵 = (Base‘𝐾)
cvrletr.l = (le‘𝐾)
cvrletr.s < = (lt‘𝐾)
cvrletr.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrnbtwn2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))

Proof of Theorem cvrnbtwn2
StepHypRef Expression
1 cvrletr.b . . . . . 6 𝐵 = (Base‘𝐾)
2 cvrletr.s . . . . . 6 < = (lt‘𝐾)
3 cvrletr.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 35227 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
543expia 1150 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
6 iman 390 . . . . 5 (((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌))
7 simpl 474 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Poset)
8 simpr3 1252 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
9 simpr2 1250 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
10 cvrletr.l . . . . . . . . . . 11 = (le‘𝐾)
1110, 2pltval 17226 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
127, 8, 9, 11syl3anc 1490 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
13 df-ne 2938 . . . . . . . . . 10 (𝑍𝑌 ↔ ¬ 𝑍 = 𝑌)
1413anbi2i 616 . . . . . . . . 9 ((𝑍 𝑌𝑍𝑌) ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))
1512, 14syl6bb 278 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1615anbi2d 622 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))))
17 anass 460 . . . . . . 7 (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1816, 17syl6rbbr 281 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
1918notbid 309 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
206, 19syl5rbb 275 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
215, 20sylibd 230 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
22213impia 1145 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌))
231, 2, 3cvrlt 35226 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
2423ex 401 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 < 𝑌))
25243adant3r3 1235 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 < 𝑌))
26253impia 1145 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27 breq2 4813 . . . 4 (𝑍 = 𝑌 → (𝑋 < 𝑍𝑋 < 𝑌))
2826, 27syl5ibrcom 238 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 < 𝑍))
291, 10posref 17217 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑌𝐵) → 𝑌 𝑌)
30293ad2antr2 1240 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌 𝑌)
31 breq1 4812 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑌𝑌 𝑌))
3230, 31syl5ibrcom 238 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 = 𝑌𝑍 𝑌))
33323adant3 1162 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3428, 33jcad 508 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍𝑍 𝑌)))
3522, 34impbid 203 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  Basecbs 16130  lecple 16221  Posetcpo 17206  ltcplt 17207  ccvr 35218
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 17194  df-poset 17212  df-plt 17224  df-covers 35222
This theorem is referenced by:  cvrval3  35369  cvrexchlem  35375
  Copyright terms: Public domain W3C validator