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Theorem cvrne 29776
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 2412 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrne.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 29765 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X
( lt `  K
) Y )
5 eqid 2412 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
65, 2pltval 14380 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( lt
`  K ) Y  <-> 
( X ( le
`  K ) Y  /\  X  =/=  Y
) ) )
76simplbda 608 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X ( lt `  K ) Y )  ->  X  =/=  Y )
84, 7syldan 457 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421   Basecbs 13432   lecple 13499   ltcplt 14361    <o ccvr 29757
This theorem is referenced by:  cvrnrefN  29777  cvrcmp  29778  cdleme3b  30723  cdleme3c  30724  cdleme7e  30741
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-plt 14378  df-covers 29761
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