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Theorem cvrntr 39408
Description: The covers relation is not transitive. (cvntr 32321 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 2735 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
3 cvrntr.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrlt 39252 . . . 4 (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌)
54ex 412 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋(lt‘𝐾)𝑌))
653adant3r3 1183 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋(lt‘𝐾)𝑌))
71, 2, 3ltcvrntr 39407 . 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 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  Basecbs 17245  ltcplt 18366  ccvr 39244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-covers 39248
This theorem is referenced by:  atcvr0eq  39409  lnnat  39410
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