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Theorem cvrne 30177
Description: The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
Hypotheses
Ref Expression
cvrne.b  |-  B  =  ( Base `  K
)
cvrne.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrne  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X  =/=  Y )

Proof of Theorem cvrne
StepHypRef Expression
1 cvrne.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2442 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrne.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 30166 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X
( lt `  K
) Y )
5 eqid 2442 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
65, 2pltval 14448 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( lt
`  K ) Y  <-> 
( X ( le
`  K ) Y  /\  X  =/=  Y
) ) )
76simplbda 609 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X ( lt `  K ) Y )  ->  X  =/=  Y )
84, 7syldan 458 1  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X  =/=  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   class class class wbr 4237   ` cfv 5483   Basecbs 13500   lecple 13567   ltcplt 14429    <o ccvr 30158
This theorem is referenced by:  cvrnrefN  30178  cvrcmp  30179  cdleme3b  31124  cdleme3c  31125  cdleme7e  31142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-plt 14446  df-covers 30162
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