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Theorem cvrnbtwn 36897
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 30213 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 36895 . . . 4 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
543adant3r3 1185 . . 3 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
6 ralnex 3148 . . . . . . 7 (∀𝑧𝐵 ¬ (𝑋 < 𝑧𝑧 < 𝑌) ↔ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))
7 breq2 5031 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑋 < 𝑧𝑋 < 𝑍))
8 breq1 5030 . . . . . . . . . 10 (𝑧 = 𝑍 → (𝑧 < 𝑌𝑍 < 𝑌))
97, 8anbi12d 634 . . . . . . . . 9 (𝑧 = 𝑍 → ((𝑋 < 𝑧𝑧 < 𝑌) ↔ (𝑋 < 𝑍𝑍 < 𝑌)))
109notbid 321 . . . . . . . 8 (𝑧 = 𝑍 → (¬ (𝑋 < 𝑧𝑧 < 𝑌) ↔ ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1110rspcv 3519 . . . . . . 7 (𝑍𝐵 → (∀𝑧𝐵 ¬ (𝑋 < 𝑧𝑧 < 𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
126, 11syl5bir 246 . . . . . 6 (𝑍𝐵 → (¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1312adantld 494 . . . . 5 (𝑍𝐵 → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
14133ad2ant3 1136 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
1514adantl 485 . . 3 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
165, 15sylbid 243 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍𝑍 < 𝑌)))
17163impia 1118 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  wrex 3054   class class class wbr 5027  cfv 6333  Basecbs 16579  ltcplt 17660  ccvr 36888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-covers 36892
This theorem is referenced by:  cvrnbtwn2  36901  cvrnbtwn3  36902  cvrnbtwn4  36905  ltltncvr  37049
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