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Theorem cvrnle 36296
Description: The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrle.b 𝐵 = (Base‘𝐾)
cvrle.l = (le‘𝐾)
cvrle.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrnle (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 𝑋)

Proof of Theorem cvrnle
StepHypRef Expression
1 cvrle.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2818 . . 3 (lt‘𝐾) = (lt‘𝐾)
3 cvrle.c . . 3 𝐶 = ( ⋖ ‘𝐾)
41, 2, 3cvrlt 36286 . 2 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌)
5 cvrle.l . . 3 = (le‘𝐾)
61, 5, 2pltnle 17564 . 2 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋(lt‘𝐾)𝑌) → ¬ 𝑌 𝑋)
74, 6syldan 591 1 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  Posetcpo 17538  ltcplt 17539  ccvr 36278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-proset 17526  df-poset 17544  df-plt 17556  df-covers 36282
This theorem is referenced by:  atnle0  36325  dih1  38302
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