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Theorem cvrnbtwn2 36293
Description: The covers relation implies no in-betweenness. (cvnbtwn2 29992 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 36289 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
543expia 1113 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
6 iman 402 . . . . 5 (((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌))
7 simpl 483 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Poset)
8 simpr3 1188 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
9 simpr2 1187 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
10 cvrletr.l . . . . . . . . . . 11 = (le‘𝐾)
1110, 2pltval 17560 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑍𝐵𝑌𝐵) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
127, 8, 9, 11syl3anc 1363 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌𝑍𝑌)))
13 df-ne 3017 . . . . . . . . . 10 (𝑍𝑌 ↔ ¬ 𝑍 = 𝑌)
1413anbi2i 622 . . . . . . . . 9 ((𝑍 𝑌𝑍𝑌) ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))
1512, 14syl6bb 288 . . . . . . . 8 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1615anbi2d 628 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌))))
17 anass 469 . . . . . . 7 (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 𝑌 ∧ ¬ 𝑍 = 𝑌)))
1816, 17syl6rbbr 291 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
1918notbid 319 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ ((𝑋 < 𝑍𝑍 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
206, 19syl5rbb 285 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
215, 20sylibd 240 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌)))
22213impia 1109 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) → 𝑍 = 𝑌))
231, 2, 3cvrlt 36288 . . . . . . 7 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
2423ex 413 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 < 𝑌))
25243adant3r3 1176 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋 < 𝑌))
26253impia 1109 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27 breq2 5062 . . . 4 (𝑍 = 𝑌 → (𝑋 < 𝑍𝑋 < 𝑌))
2826, 27syl5ibrcom 248 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑋 < 𝑍))
291, 10posref 17551 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑌𝐵) → 𝑌 𝑌)
30293ad2antr2 1181 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌 𝑌)
31 breq1 5061 . . . . 5 (𝑍 = 𝑌 → (𝑍 𝑌𝑌 𝑌))
3230, 31syl5ibrcom 248 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 = 𝑌𝑍 𝑌))
33323adant3 1124 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌𝑍 𝑌))
3428, 33jcad 513 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍𝑍 𝑌)))
3522, 34impbid 213 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  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:  cvrval3  36431  cvrexchlem  36437
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