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Theorem lcvnbtwn 29215
Description: The covers relation implies no in-betweenness. (cvnbtwn 22866 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 29211 . . . 4  |-  ( ph  ->  ( R C T  <-> 
( R  C.  T  /\  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) ) ) )
81, 7mpbid 201 . . 3  |-  ( ph  ->  ( R  C.  T  /\  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) ) )
98simprd 449 . 2  |-  ( ph  ->  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) )
10 lcvnbtwn.u . . 3  |-  ( ph  ->  U  e.  S )
11 psseq2 3264 . . . . 5  |-  ( u  =  U  ->  ( R  C.  u  <->  R  C.  U ) )
12 psseq1 3263 . . . . 5  |-  ( u  =  U  ->  (
u  C.  T  <->  U  C.  T ) )
1311, 12anbi12d 691 . . . 4  |-  ( u  =  U  ->  (
( R  C.  u  /\  u  C.  T )  <-> 
( R  C.  U  /\  U  C.  T ) ) )
1413rspcev 2884 . . 3  |-  ( ( U  e.  S  /\  ( R  C.  U  /\  U  C.  T ) )  ->  E. u  e.  S  ( R  C.  u  /\  u  C.  T ) )
1510, 14sylan 457 . 2  |-  ( (
ph  /\  ( R  C.  U  /\  U  C.  T ) )  ->  E. u  e.  S  ( R  C.  u  /\  u  C.  T ) )
169, 15mtand 640 1  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C. wpss 3153   class class class wbr 4023   ` cfv 5255   LSubSpclss 15689    <oLL clcv 29208
This theorem is referenced by:  lcvntr  29216  lcvnbtwn2  29217  lcvnbtwn3  29218
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-pss 3168  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-lcv 29209
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