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Theorem cvrnbtwn 38654
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 32048 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 38652 . . . 4 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
543adant3r3 1181 . . 3 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
6 ralnex 3066 . . . . . . 7 (∀𝑧𝐵 ¬ (𝑋 < 𝑧𝑧 < 𝑌) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))
7 breq2 5145 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋 < 𝑧𝑋 < 𝑍))
8 breq1 5144 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑧 < 𝑌𝑍 < 𝑌))
97, 8anbi12d 630 . . . . . . . . 9 (𝑧 = 𝑍 → ((𝑋 < 𝑧𝑧 < 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
109notbid 318 . . . . . . . 8 (𝑧 = 𝑍 → (¬ (𝑋 < 𝑧𝑧 < 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1110rspcv 3602 . . . . . . 7 (𝑍𝐵 → (∀𝑧𝐵 ¬ (𝑋 < 𝑧𝑧 < 𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
126, 11biimtrrid 242 . . . . . 6 (𝑍𝐵 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1312adantld 490 . . . . 5 (𝑍𝐵 → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
14133ad2ant3 1132 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1514adantl 481 . . 3 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
165, 15sylbid 239 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
17163impia 1114 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wrex 3064   class class class wbr 5141  cfv 6537  Basecbs 17153  ltcplt 18273  ccvr 38645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-covers 38649
This theorem is referenced by:  cvrnbtwn2  38658  cvrnbtwn3  38659  cvrnbtwn4  38662  ltltncvr  38807
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