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Theorem lcvnbtwn 29284
Description: The covers relation implies no in-betweenness. (cvnbtwn 22980 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 29280 . . . 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 3340 . . . . 5  |-  ( u  =  U  ->  ( R  C.  u  <->  R  C.  U ) )
12 psseq1 3339 . . . . 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 2960 . . 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 1642    e. wcel 1710   E.wrex 2620    C. wpss 3229   class class class wbr 4104   ` cfv 5337   LSubSpclss 15788    <oLL clcv 29277
This theorem is referenced by:  lcvntr  29285  lcvnbtwn2  29286  lcvnbtwn3  29287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-lcv 29278
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