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Theorem cvrntr 37447
Description: The covers relation is not transitive. (cvntr 30662 analog.) (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
cvrntr.b 𝐵 = (Base‘𝐾)
cvrntr.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrntr ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))

Proof of Theorem cvrntr
StepHypRef Expression
1 cvrntr.b . . . . 5 𝐵 = (Base‘𝐾)
2 eqid 2738 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
3 cvrntr.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrlt 37292 . . . 4 (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌)
54ex 413 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋(lt‘𝐾)𝑌))
653adant3r3 1183 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋(lt‘𝐾)𝑌))
71, 2, 3ltcvrntr 37446 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(lt‘𝐾)𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
86, 7syland 603 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106   class class class wbr 5073  cfv 6426  Basecbs 16922  ltcplt 18036  ccvr 37284
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 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
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 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-covers 37288
This theorem is referenced by:  atcvr0eq  37448  lnnat  37449
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