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Theorem cvrne 29530
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 2366 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrne.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 29519 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X
( lt `  K
) Y )
5 eqid 2366 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
65, 2pltval 14304 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( lt
`  K ) Y  <-> 
( X ( le
`  K ) Y  /\  X  =/=  Y
) ) )
76simplbda 607 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X ( lt `  K ) Y )  ->  X  =/=  Y )
84, 7syldan 456 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 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358   Basecbs 13356   lecple 13423   ltcplt 14285    <o ccvr 29511
This theorem is referenced by:  cvrnrefN  29531  cvrcmp  29532  cdleme3b  30477  cdleme3c  30478  cdleme7e  30495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-plt 14302  df-covers 29515
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