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Theorem lcvnbtwn2 29825
Description: The covers relation implies no in-betweenness. (cvnbtwn2 23790 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 29823 . . 3  |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T
) )
11 iman 414 . . . 4  |-  ( ( ( R  C.  U  /\  U  C_  T )  ->  U  =  T )  <->  -.  ( ( R  C.  U  /\  U  C_  T )  /\  -.  U  =  T )
)
12 anass 631 . . . . . 6  |-  ( ( ( R  C.  U  /\  U  C_  T )  /\  -.  U  =  T )  <->  ( R  C.  U  /\  ( U  C_  T  /\  -.  U  =  T )
) )
13 dfpss2 3432 . . . . . . 7  |-  ( U 
C.  T  <->  ( U  C_  T  /\  -.  U  =  T ) )
1413anbi2i 676 . . . . . 6  |-  ( ( R  C.  U  /\  U  C.  T )  <->  ( R  C.  U  /\  ( U  C_  T  /\  -.  U  =  T )
) )
1512, 14bitr4i 244 . . . . 5  |-  ( ( ( R  C.  U  /\  U  C_  T )  /\  -.  U  =  T )  <->  ( R  C.  U  /\  U  C.  T ) )
1615notbii 288 . . . 4  |-  ( -.  ( ( R  C.  U  /\  U  C_  T
)  /\  -.  U  =  T )  <->  -.  ( R  C.  U  /\  U  C.  T ) )
1711, 16bitr2i 242 . . 3  |-  ( -.  ( R  C.  U  /\  U  C.  T )  <-> 
( ( R  C.  U  /\  U  C_  T
)  ->  U  =  T ) )
1810, 17sylib 189 . 2  |-  ( ph  ->  ( ( R  C.  U  /\  U  C_  T
)  ->  U  =  T ) )
191, 2, 18mp2and 661 1  |-  ( ph  ->  U  =  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320    C. wpss 3321   class class class wbr 4212   ` cfv 5454   LSubSpclss 16008    <oLL clcv 29816
This theorem is referenced by:  lcvat  29828  lsatexch  29841
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-lcv 29817
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