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Theorem lcvnbtwn3 32563
Description: The covers relation implies no in-betweenness. (cvnbtwn3 27939 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 32560 . . 3  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
11 iman 425 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  R  =  U )  <->  -.  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
12 eqcom 2431 . . . . 5  |-  ( U  =  R  <->  R  =  U )
1312imbi2i 313 . . . 4  |-  ( ( ( R  C_  U  /\  U  C.  T )  ->  U  =  R )  <->  ( ( R 
C_  U  /\  U  C.  T )  ->  R  =  U ) )
14 dfpss2 3550 . . . . . . 7  |-  ( R 
C.  U  <->  ( R  C_  U  /\  -.  R  =  U ) )
1514anbi1i 699 . . . . . 6  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T ) )
16 an32 805 . . . . . 6  |-  ( ( ( R  C_  U  /\  -.  R  =  U )  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1715, 16bitri 252 . . . . 5  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( ( R  C_  U  /\  U  C.  T )  /\  -.  R  =  U )
)
1817notbii 297 . . . 4  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <->  -.  ( ( R  C_  U  /\  U  C.  T
)  /\  -.  R  =  U ) )
1911, 13, 183bitr4ri 281 . . 3  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <-> 
( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
2010, 19sylib 199 . 2  |-  ( ph  ->  ( ( R  C_  U  /\  U  C.  T
)  ->  U  =  R ) )
211, 2, 20mp2and 683 1  |-  ( ph  ->  U  =  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436    C. wpss 3437   class class class wbr 4423   ` cfv 5601   LSubSpclss 18154    <oLL clcv 32553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-lcv 32554
This theorem is referenced by:  lsatcveq0  32567  lsatcvatlem  32584
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