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Theorem cvrnbtwn3 36294
Description: The covers relation implies no in-betweenness. (cvnbtwn3 29993 analog.) (Contributed by NM, 4-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
cvrnbtwn3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

Proof of Theorem cvrnbtwn3
StepHypRef Expression
1 cvrletr.b . . . 4 𝐵 = (Base‘𝐾)
2 cvrletr.s . . . 4 < = (lt‘𝐾)
3 cvrletr.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrnbtwn 36289 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
5 cvrletr.l . . . . . . . . 9 = (le‘𝐾)
65, 2pltval 17560 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
763adant3r2 1175 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3 1124 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
98anbi1d 629 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
109notbid 319 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
11 an32 642 . . . . . . 7 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍))
12 df-ne 3017 . . . . . . . 8 (𝑋𝑍 ↔ ¬ 𝑋 = 𝑍)
1312anbi2i 622 . . . . . . 7 (((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1411, 13bitri 276 . . . . . 6 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1514notbii 321 . . . . 5 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16 iman 402 . . . . 5 (((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1715, 16bitr4i 279 . . . 4 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
1810, 17syl6bb 288 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍)))
194, 18mpbid 233 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
201, 5posref 17551 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
21 breq2 5062 . . . . . 6 (𝑋 = 𝑍 → (𝑋 𝑋𝑋 𝑍))
2220, 21syl5ibcom 246 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑍𝑋 𝑍))
23223ad2antr1 1180 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍𝑋 𝑍))
24233adant3 1124 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
25 simp1 1128 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26 simp21 1198 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
27 simp22 1199 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
28 simp3 1130 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
291, 2, 3cvrlt 36288 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
3025, 26, 27, 28, 29syl31anc 1365 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31 breq1 5061 . . . 4 (𝑋 = 𝑍 → (𝑋 < 𝑌𝑍 < 𝑌))
3230, 31syl5ibcom 246 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 < 𝑌))
3324, 32jcad 513 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 𝑍𝑍 < 𝑌)))
3419, 33impbid 213 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))
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:  atcvreq0  36332  cvratlem  36439
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