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Theorem cvrnbtwn 37285
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 30648 analog.) (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrfval.b 𝐵 = (Base‘𝐾)
cvrfval.s < = (lt‘𝐾)
cvrfval.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrnbtwn ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))

Proof of Theorem cvrnbtwn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cvrfval.b . . . . 5 𝐵 = (Base‘𝐾)
2 cvrfval.s . . . . 5 < = (lt‘𝐾)
3 cvrfval.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrval 37283 . . . 4 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
543adant3r3 1183 . . 3 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
6 ralnex 3167 . . . . . . 7 (∀𝑧𝐵 ¬ (𝑋 < 𝑧𝑧 < 𝑌) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))
7 breq2 5078 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋 < 𝑧𝑋 < 𝑍))
8 breq1 5077 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑧 < 𝑌𝑍 < 𝑌))
97, 8anbi12d 631 . . . . . . . . 9 (𝑧 = 𝑍 → ((𝑋 < 𝑧𝑧 < 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
109notbid 318 . . . . . . . 8 (𝑧 = 𝑍 → (¬ (𝑋 < 𝑧𝑧 < 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1110rspcv 3557 . . . . . . 7 (𝑍𝐵 → (∀𝑧𝐵 ¬ (𝑋 < 𝑧𝑧 < 𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
126, 11syl5bir 242 . . . . . 6 (𝑍𝐵 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1312adantld 491 . . . . 5 (𝑍𝐵 → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
14133ad2ant3 1134 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1514adantl 482 . . 3 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
165, 15sylbid 239 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
17163impia 1116 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065   class class class wbr 5074  cfv 6433  Basecbs 16912  ltcplt 18026  ccvr 37276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-covers 37280
This theorem is referenced by:  cvrnbtwn2  37289  cvrnbtwn3  37290  cvrnbtwn4  37293  ltltncvr  37437
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