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Theorem cvrnbtwn3 33581
 Description: The covers relation implies no in-betweenness. (cvnbtwn3 28531 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 33576 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
5 cvrletr.l . . . . . . . . 9 = (le‘𝐾)
65, 2pltval 16783 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
763adant3r2 1267 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
873adant3 1074 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 𝑍𝑋𝑍)))
98anbi1d 737 . . . . 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 835 . . . . . . 7 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍))
12 df-ne 2782 . . . . . . . 8 (𝑋𝑍 ↔ ¬ 𝑋 = 𝑍)
1312anbi2i 726 . . . . . . 7 (((𝑋 𝑍𝑍 < 𝑌) ∧ 𝑋𝑍) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1411, 13bitri 263 . . . . . 6 (((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1514notbii 309 . . . . 5 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16 iman 439 . . . . 5 (((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 𝑍𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
1715, 16bitr4i 266 . . . 4 (¬ ((𝑋 𝑍𝑋𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
1810, 17syl6bb 275 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍𝑍 < 𝑌) ↔ ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍)))
194, 18mpbid 221 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) → 𝑋 = 𝑍))
201, 5posref 16774 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
21 breq2 4587 . . . . . 6 (𝑋 = 𝑍 → (𝑋 𝑋𝑋 𝑍))
2220, 21syl5ibcom 234 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 = 𝑍𝑋 𝑍))
23223ad2antr1 1219 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍𝑋 𝑍))
24233adant3 1074 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑋 𝑍))
25 simp1 1054 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26 simp21 1087 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
27 simp22 1088 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
28 simp3 1056 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
291, 2, 3cvrlt 33575 . . . . 5 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
3025, 26, 27, 28, 29syl31anc 1321 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31 breq1 4586 . . . 4 (𝑋 = 𝑍 → (𝑋 < 𝑌𝑍 < 𝑌))
3230, 31syl5ibcom 234 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍𝑍 < 𝑌))
3324, 32jcad 554 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 𝑍𝑍 < 𝑌)))
3419, 33impbid 201 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764   ⋖ ccvr 33567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781  df-covers 33571 This theorem is referenced by:  atcvreq0  33619  cvratlem  33725
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