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Theorem lcvnbtwn2 29217
Description: The covers relation implies no in-betweenness. (cvnbtwn2 22867 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 )
lcvnbtwn2.p  |-  ( ph  ->  R  C.  U )
lcvnbtwn2.q  |-  ( ph  ->  U  C_  T )
Assertion
Ref Expression
lcvnbtwn2  |-  ( ph  ->  U  =  T )

Proof of Theorem lcvnbtwn2
StepHypRef Expression
1 lcvnbtwn2.p . 2  |-  ( ph  ->  R  C.  U )
2 lcvnbtwn2.q . 2  |-  ( ph  ->  U  C_  T )
3 lcvnbtwn.s . . . 4  |-  S  =  ( LSubSp `  W )
4 lcvnbtwn.c . . . 4  |-  C  =  (  <oLL  `  W )
5 lcvnbtwn.w . . . 4  |-  ( ph  ->  W  e.  X )
6 lcvnbtwn.r . . . 4  |-  ( ph  ->  R  e.  S )
7 lcvnbtwn.t . . . 4  |-  ( ph  ->  T  e.  S )
8 lcvnbtwn.u . . . 4  |-  ( ph  ->  U  e.  S )
9 lcvnbtwn.d . . . 4  |-  ( ph  ->  R C T )
103, 4, 5, 6, 7, 8, 9lcvnbtwn 29215 . . 3  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
11 iman 413 . . . 4  |-  ( ( ( R  C.  U  /\  U  C_  T )  ->  U  =  T )  <->  -.  ( ( R  C.  U  /\  U  C_  T )  /\  -.  U  =  T )
)
12 anass 630 . . . . . 6  |-  ( ( ( R  C.  U  /\  U  C_  T )  /\  -.  U  =  T )  <->  ( R  C.  U  /\  ( U  C_  T  /\  -.  U  =  T )
) )
13 dfpss2 3261 . . . . . . 7  |-  ( U 
C.  T  <->  ( U  C_  T  /\  -.  U  =  T ) )
1413anbi2i 675 . . . . . 6  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( R  C.  U  /\  ( U  C_  T  /\  -.  U  =  T )
) )
1512, 14bitr4i 243 . . . . 5  |-  ( ( ( R  C.  U  /\  U  C_  T )  /\  -.  U  =  T )  <->  ( R  C.  U  /\  U  C.  T ) )
1615notbii 287 . . . 4  |-  ( -.  ( ( R  C.  U  /\  U  C_  T
)  /\  -.  U  =  T )  <->  -.  ( R  C.  U  /\  U  C.  T ) )
1711, 16bitr2i 241 . . 3  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <-> 
( ( R  C.  U  /\  U  C_  T
)  ->  U  =  T ) )
1810, 17sylib 188 . 2  |-  ( ph  ->  ( ( R  C.  U  /\  U  C_  T
)  ->  U  =  T ) )
191, 2, 18mp2and 660 1  |-  ( ph  ->  U  =  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152    C. wpss 3153   class class class wbr 4023   ` cfv 5255   LSubSpclss 15689    <oLL clcv 29208
This theorem is referenced by:  lcvat  29220  lsatexch  29233
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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