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Theorem cvrnbtwn3 34040
Description: The covers relation implies no in-betweenness. (cvnbtwn3 28993 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 34035 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
5 cvrletr.l . . . . . . . . 9 = (le‘𝐾)
65, 2pltval 16881 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
763adant3r2 1272 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3 1079 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
98anbi1d 740 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
109notbid 308 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
11 an32 838 . . . . . . 7 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍))
12 df-ne 2791 . . . . . . . 8 (𝑋𝑍 ↔ ¬ 𝑋 = 𝑍)
1312anbi2i 729 . . . . . . 7 (((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1411, 13bitri 264 . . . . . 6 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1514notbii 310 . . . . 5 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16 iman 440 . . . . 5 (((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1715, 16bitr4i 267 . . . 4 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
1810, 17syl6bb 276 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍)))
194, 18mpbid 222 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
201, 5posref 16872 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
21 breq2 4617 . . . . . 6 (𝑋 = 𝑍 → (𝑋 𝑋𝑋 𝑍))
2220, 21syl5ibcom 235 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑍𝑋 𝑍))
23223ad2antr1 1224 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍𝑋 𝑍))
24233adant3 1079 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
25 simp1 1059 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26 simp21 1092 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
27 simp22 1093 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
28 simp3 1061 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
291, 2, 3cvrlt 34034 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
3025, 26, 27, 28, 29syl31anc 1326 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31 breq1 4616 . . . 4 (𝑋 = 𝑍 → (𝑋 < 𝑌𝑍 < 𝑌))
3230, 31syl5ibcom 235 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 < 𝑌))
3324, 32jcad 555 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 𝑍𝑍 < 𝑌)))
3419, 33impbid 202 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  Posetcpo 16861  ltcplt 16862  ccvr 34026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-preset 16849  df-poset 16867  df-plt 16879  df-covers 34030
This theorem is referenced by:  atcvreq0  34078  cvratlem  34184
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