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Theorem cvrnle 29763
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 2404 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrle.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 29753 . 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 14378 . 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 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353    <o ccvr 29745
This theorem is referenced by:  atnle0  29792  dih1  31769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-poset 14358  df-plt 14370  df-covers 29749
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