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Theorem ltcvrntr 37022
 Description: Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
ltltncvr.b 𝐵 = (Base‘𝐾)
ltltncvr.s < = (lt‘𝐾)
ltltncvr.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
ltcvrntr ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Proof of Theorem ltcvrntr
StepHypRef Expression
1 ltltncvr.b . . . . 5 𝐵 = (Base‘𝐾)
2 ltltncvr.s . . . . 5 < = (lt‘𝐾)
3 ltltncvr.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrlt 36868 . . . 4 (((𝐾𝐴𝑌𝐵𝑍𝐵) ∧ 𝑌𝐶𝑍) → 𝑌 < 𝑍)
54ex 416 . . 3 ((𝐾𝐴𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍𝑌 < 𝑍))
653adant3r1 1179 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍𝑌 < 𝑍))
71, 2, 3ltltncvr 37021 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))
86, 7sylan2d 607 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5032  ‘cfv 6335  Basecbs 16541  ltcplt 17617   ⋖ ccvr 36860 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-covers 36864 This theorem is referenced by:  cvrntr  37023
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