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Theorem cvrnbtwn2 39231
Description: The covers relation implies no in-betweenness. (cvnbtwn2 32319 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 39227 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
543expia 1121 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
6 iman 401 . . . . 5 (((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌))
7 anass 468 . . . . . . 7 (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
8 simpl 482 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Poset)
9 simpr3 1196 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
10 simpr2 1195 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
11 cvrletr.l . . . . . . . . . . 11 = (le‘𝐾)
1211, 2pltval 18402 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
138, 9, 10, 12syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
14 df-ne 2947 . . . . . . . . . 10 (𝑍𝑌 ↔ ¬ 𝑍 = 𝑌)
1514anbi2i 622 . . . . . . . . 9 ((𝑍 𝑌𝑍𝑌) ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))
1613, 15bitrdi 287 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1716anbi2d 629 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))))
187, 17bitr4id 290 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
1918notbid 318 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
206, 19bitr2id 284 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
215, 20sylibd 239 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
22213impia 1117 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌))
231, 2, 3cvrlt 39226 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
2423ex 412 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 < 𝑌))
25243adant3r3 1184 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 < 𝑌))
26253impia 1117 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27 breq2 5170 . . . 4 (𝑍 = 𝑌 → (𝑋 < 𝑍𝑋 < 𝑌))
2826, 27syl5ibrcom 247 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 < 𝑍))
291, 11posref 18388 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑌𝐵) → 𝑌 𝑌)
30293ad2antr2 1189 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌 𝑌)
31 breq1 5169 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑌𝑌 𝑌))
3230, 31syl5ibrcom 247 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 = 𝑌𝑍 𝑌))
33323adant3 1132 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3428, 33jcad 512 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍𝑍 𝑌)))
3522, 34impbid 212 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wne 2946   class class class wbr 5166  cfv 6573  Basecbs 17258  lecple 17318  Posetcpo 18377  ltcplt 18378  ccvr 39218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-proset 18365  df-poset 18383  df-plt 18400  df-covers 39222
This theorem is referenced by:  cvrval3  39370  cvrexchlem  39376
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