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Theorem lcvnbtwn 29897
Description: The covers relation implies no in-betweenness. (cvnbtwn 23794 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvnbtwn.s  |-  S  =  ( LSubSp `  W )
lcvnbtwn.c  |-  C  =  (  <oLL  `  W )
lcvnbtwn.w  |-  ( ph  ->  W  e.  X )
lcvnbtwn.r  |-  ( ph  ->  R  e.  S )
lcvnbtwn.t  |-  ( ph  ->  T  e.  S )
lcvnbtwn.u  |-  ( ph  ->  U  e.  S )
lcvnbtwn.d  |-  ( ph  ->  R C T )
Assertion
Ref Expression
lcvnbtwn  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )

Proof of Theorem lcvnbtwn
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 lcvnbtwn.d . . . 4  |-  ( ph  ->  R C T )
2 lcvnbtwn.s . . . . 5  |-  S  =  ( LSubSp `  W )
3 lcvnbtwn.c . . . . 5  |-  C  =  (  <oLL  `  W )
4 lcvnbtwn.w . . . . 5  |-  ( ph  ->  W  e.  X )
5 lcvnbtwn.r . . . . 5  |-  ( ph  ->  R  e.  S )
6 lcvnbtwn.t . . . . 5  |-  ( ph  ->  T  e.  S )
72, 3, 4, 5, 6lcvbr 29893 . . . 4  |-  ( ph  ->  ( R C T  <-> 
( R  C.  T  /\  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) ) ) )
81, 7mpbid 203 . . 3  |-  ( ph  ->  ( R  C.  T  /\  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) ) )
98simprd 451 . 2  |-  ( ph  ->  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) )
10 lcvnbtwn.u . . 3  |-  ( ph  ->  U  e.  S )
11 psseq2 3437 . . . . 5  |-  ( u  =  U  ->  ( R  C.  u  <->  R  C.  U ) )
12 psseq1 3436 . . . . 5  |-  ( u  =  U  ->  (
u  C.  T  <->  U  C.  T ) )
1311, 12anbi12d 693 . . . 4  |-  ( u  =  U  ->  (
( R  C.  u  /\  u  C.  T )  <-> 
( R  C.  U  /\  U  C.  T ) ) )
1413rspcev 3054 . . 3  |-  ( ( U  e.  S  /\  ( R  C.  U  /\  U  C.  T ) )  ->  E. u  e.  S  ( R  C.  u  /\  u  C.  T ) )
1510, 14sylan 459 . 2  |-  ( (
ph  /\  ( R  C.  U  /\  U  C.  T ) )  ->  E. u  e.  S  ( R  C.  u  /\  u  C.  T ) )
169, 15mtand 642 1  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    C. wpss 3323   class class class wbr 4215   ` cfv 5457   LSubSpclss 16013    <oLL clcv 29890
This theorem is referenced by:  lcvntr  29898  lcvnbtwn2  29899  lcvnbtwn3  29900
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 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-lcv 29891
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