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Theorem lcvnbtwn3 29665
Description: The covers relation implies no in-betweenness. (cvnbtwn3 23779 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 )
lcvnbtwn3.p  |-  ( ph  ->  R  C_  U )
lcvnbtwn3.q  |-  ( ph  ->  U  C.  T )
Assertion
Ref Expression
lcvnbtwn3  |-  ( ph  ->  U  =  R )

Proof of Theorem lcvnbtwn3
StepHypRef Expression
1 lcvnbtwn3.p . 2  |-  ( ph  ->  R  C_  U )
2 lcvnbtwn3.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 29662 . . 3  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
11 iman 414 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  R  =  U )  <->  -.  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
12 eqcom 2437 . . . . 5  |-  ( U  =  R  <->  R  =  U )
1312imbi2i 304 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  U  =  R )  <->  ( ( R 
C_  U  /\  U  C.  T )  ->  R  =  U ) )
14 dfpss2 3424 . . . . . . 7  |-  ( R 
C.  U  <->  ( R  C_  U  /\  -.  R  =  U ) )
1514anbi1i 677 . . . . . 6  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T ) )
16 an32 774 . . . . . 6  |-  ( ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1715, 16bitri 241 . . . . 5  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1817notbii 288 . . . 4  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <->  -.  ( ( R  C_  U  /\  U  C.  T
)  /\  -.  R  =  U ) )
1911, 13, 183bitr4ri 270 . . 3  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <-> 
( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
2010, 19sylib 189 . 2  |-  ( ph  ->  ( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
211, 2, 20mp2and 661 1  |-  ( ph  ->  U  =  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312    C. wpss 3313   class class class wbr 4204   ` cfv 5445   LSubSpclss 15996    <oLL clcv 29655
This theorem is referenced by:  lsatcveq0  29669  lsatcvatlem  29686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-lcv 29656
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