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Theorem cvrntr 36621
Description: The covers relation is not transitive. (cvntr 30064 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 2824 . . . . 5 (lt‘𝐾) = (lt‘𝐾)
3 cvrntr.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrlt 36466 . . . 4 (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌)
54ex 416 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋(lt‘𝐾)𝑌))
653adant3r3 1181 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋(lt‘𝐾)𝑌))
71, 2, 3ltcvrntr 36620 . 2 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(lt‘𝐾)𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
86, 7syland 605 1 ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5047  cfv 6336  Basecbs 16472  ltcplt 17540  ccvr 36458
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-covers 36462
This theorem is referenced by:  atcvr0eq  36622  lnnat  36623
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