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Theorem ltcvrntr 38164
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 38009 . . . 4 (((𝐾𝐴𝑌𝐵𝑍𝐵) ∧ 𝑌𝐶𝑍) → 𝑌 < 𝑍)
54ex 413 . . 3 ((𝐾𝐴𝑌𝐵𝑍𝐵) → (𝑌𝐶𝑍𝑌 < 𝑍))
653adant3r1 1182 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌𝐶𝑍𝑌 < 𝑍))
71, 2, 3ltltncvr 38163 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍))
86, 7sylan2d 605 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5142  cfv 6533  Basecbs 17128  ltcplt 18245  ccvr 38001
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541  df-covers 38005
This theorem is referenced by:  cvrntr  38165
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