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Theorem cvrnbtwn2 36571
Description: The covers relation implies no in-betweenness. (cvnbtwn2 30070 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 36567 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
543expia 1118 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
6 iman 405 . . . . 5 (((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌))
7 anass 472 . . . . . . 7 (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
8 simpl 486 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Poset)
9 simpr3 1193 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
10 simpr2 1192 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
11 cvrletr.l . . . . . . . . . . 11 = (le‘𝐾)
1211, 2pltval 17562 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
138, 9, 10, 12syl3anc 1368 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
14 df-ne 2988 . . . . . . . . . 10 (𝑍𝑌 ↔ ¬ 𝑍 = 𝑌)
1514anbi2i 625 . . . . . . . . 9 ((𝑍 𝑌𝑍𝑌) ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))
1613, 15syl6bb 290 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1716anbi2d 631 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))))
187, 17bitr4id 293 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
1918notbid 321 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
206, 19syl5rbb 287 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
215, 20sylibd 242 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
22213impia 1114 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌))
231, 2, 3cvrlt 36566 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
2423ex 416 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 < 𝑌))
25243adant3r3 1181 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 < 𝑌))
26253impia 1114 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27 breq2 5034 . . . 4 (𝑍 = 𝑌 → (𝑋 < 𝑍𝑋 < 𝑌))
2826, 27syl5ibrcom 250 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 < 𝑍))
291, 11posref 17553 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑌𝐵) → 𝑌 𝑌)
30293ad2antr2 1186 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌 𝑌)
31 breq1 5033 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑌𝑌 𝑌))
3230, 31syl5ibrcom 250 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 = 𝑌𝑍 𝑌))
33323adant3 1129 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3428, 33jcad 516 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍𝑍 𝑌)))
3522, 34impbid 215 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987   class class class wbr 5030  cfv 6324  Basecbs 16475  lecple 16564  Posetcpo 17542  ltcplt 17543  ccvr 36558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-proset 17530  df-poset 17548  df-plt 17560  df-covers 36562
This theorem is referenced by:  cvrval3  36709  cvrexchlem  36715
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