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Theorem cvrnbtwn3 36484
Description: The covers relation implies no in-betweenness. (cvnbtwn3 30069 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 36479 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
5 cvrletr.l . . . . . . . . 9 = (le‘𝐾)
65, 2pltval 17568 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
763adant3r2 1180 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3 1129 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
98anbi1d 632 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
109notbid 321 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌)))
11 an32 645 . . . . . . 7 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍))
12 df-ne 3015 . . . . . . . 8 (𝑋𝑍 ↔ ¬ 𝑋 = 𝑍)
1312anbi2i 625 . . . . . . 7 (((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1411, 13bitri 278 . . . . . 6 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1514notbii 323 . . . . 5 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16 iman 405 . . . . 5 (((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1715, 16bitr4i 281 . . . 4 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
1810, 17syl6bb 290 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍)))
194, 18mpbid 235 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
201, 5posref 17559 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
21 breq2 5057 . . . . . 6 (𝑋 = 𝑍 → (𝑋 𝑋𝑋 𝑍))
2220, 21syl5ibcom 248 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑍𝑋 𝑍))
23223ad2antr1 1185 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍𝑋 𝑍))
24233adant3 1129 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
25 simp1 1133 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26 simp21 1203 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
27 simp22 1204 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
28 simp3 1135 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
291, 2, 3cvrlt 36478 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
3025, 26, 27, 28, 29syl31anc 1370 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31 breq1 5056 . . . 4 (𝑋 = 𝑍 → (𝑋 < 𝑌𝑍 < 𝑌))
3230, 31syl5ibcom 248 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 < 𝑌))
3324, 32jcad 516 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 𝑍𝑍 < 𝑌)))
3419, 33impbid 215 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5053  cfv 6344  Basecbs 16481  lecple 16570  Posetcpo 17548  ltcplt 17549  ccvr 36470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-proset 17536  df-poset 17554  df-plt 17566  df-covers 36474
This theorem is referenced by:  atcvreq0  36522  cvratlem  36629
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