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Theorem cvrnbtwn3 35078
Description: The covers relation implies no in-betweenness. (cvnbtwn3 29481 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 35073 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
5 cvrletr.l . . . . . . . . 9 = (le‘𝐾)
65, 2pltval 17167 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
763adant3r2 1197 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3 1125 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
98anbi1d 607 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
109notbid 307 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
11 an32 617 . . . . . . 7 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍))
12 df-ne 2943 . . . . . . . 8 (𝑋𝑍 ↔ ¬ 𝑋 = 𝑍)
1312anbi2i 601 . . . . . . 7 (((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1411, 13bitri 264 . . . . . 6 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1514notbii 309 . . . . 5 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16 iman 388 . . . . 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 17158 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
21 breq2 4788 . . . . . 6 (𝑋 = 𝑍 → (𝑋 𝑋𝑋 𝑍))
2220, 21syl5ibcom 235 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑍𝑋 𝑍))
23223ad2antr1 1202 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍𝑋 𝑍))
24233adant3 1125 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
25 simp1 1129 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26 simp21 1247 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
27 simp22 1248 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
28 simp3 1131 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
291, 2, 3cvrlt 35072 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
3025, 26, 27, 28, 29syl31anc 1478 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31 breq1 4787 . . . 4 (𝑋 = 𝑍 → (𝑋 < 𝑌𝑍 < 𝑌))
3230, 31syl5ibcom 235 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 < 𝑌))
3324, 32jcad 496 . 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 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942   class class class wbr 4784  cfv 6031  Basecbs 16063  lecple 16155  Posetcpo 17147  ltcplt 17148  ccvr 35064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-preset 17135  df-poset 17153  df-plt 17165  df-covers 35068
This theorem is referenced by:  atcvreq0  35116  cvratlem  35222
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