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Theorem lcvnbtwn 29520
Description: The covers relation implies no in-betweenness. (cvnbtwn 23750 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 29516 . . . 4  |-  ( ph  ->  ( R C T  <-> 
( R  C.  T  /\  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) ) ) )
81, 7mpbid 202 . . 3  |-  ( ph  ->  ( R  C.  T  /\  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) ) )
98simprd 450 . 2  |-  ( ph  ->  -.  E. u  e.  S  ( R  C.  u  /\  u  C.  T
) )
10 lcvnbtwn.u . . 3  |-  ( ph  ->  U  e.  S )
11 psseq2 3403 . . . . 5  |-  ( u  =  U  ->  ( R  C.  u  <->  R  C.  U ) )
12 psseq1 3402 . . . . 5  |-  ( u  =  U  ->  (
u  C.  T  <->  U  C.  T ) )
1311, 12anbi12d 692 . . . 4  |-  ( u  =  U  ->  (
( R  C.  u  /\  u  C.  T )  <-> 
( R  C.  U  /\  U  C.  T ) ) )
1413rspcev 3020 . . 3  |-  ( ( U  e.  S  /\  ( R  C.  U  /\  U  C.  T ) )  ->  E. u  e.  S  ( R  C.  u  /\  u  C.  T ) )
1510, 14sylan 458 . 2  |-  ( (
ph  /\  ( R  C.  U  /\  U  C.  T ) )  ->  E. u  e.  S  ( R  C.  u  /\  u  C.  T ) )
169, 15mtand 641 1  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675    C. wpss 3289   class class class wbr 4180   ` cfv 5421   LSubSpclss 15971    <oLL clcv 29513
This theorem is referenced by:  lcvntr  29521  lcvnbtwn2  29522  lcvnbtwn3  29523
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-pss 3304  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-lcv 29514
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