Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrne Unicode version

Theorem cvrne 29471
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 2283 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
3 cvrne.c . . 3  |-  C  =  (  <o  `  K )
41, 2, 3cvrlt 29460 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X C Y )  ->  X
( lt `  K
) Y )
5 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
65, 2pltval 14094 . . 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 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   ltcplt 14075    <o ccvr 29452
This theorem is referenced by:  cvrnrefN  29472  cvrcmp  29473  cdleme3b  30418  cdleme3c  30419  cdleme7e  30436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-plt 14092  df-covers 29456
  Copyright terms: Public domain W3C validator