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Theorem cvrnle 30078
Description: The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
cvrle.b  |-  B  =  ( Base `  K
)
cvrle.l  |-  .<_  =  ( le `  K )
cvrle.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrnle  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  -.  Y  .<_  X )

Proof of Theorem cvrnle
StepHypRef Expression
1 cvrle.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2436 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrle.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 30068 . 2  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X ( lt
`  K ) Y )
5 cvrle.l . . 3  |-  .<_  =  ( le `  K )
61, 5, 2pltnle 14423 . 2  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X ( lt `  K ) Y )  ->  -.  Y  .<_  X )
74, 6syldan 457 1  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  -.  Y  .<_  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   ltcplt 14398    <o ccvr 30060
This theorem is referenced by:  atnle0  30107  dih1  32084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-poset 14403  df-plt 14415  df-covers 30064
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