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Theorem lcvntr 29824
Description: The covers relation is not transitive. (cvntr 23795 analog.) (Contributed by NM, 10-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 )
lcvntr.p  |-  ( ph  ->  T C U )
Assertion
Ref Expression
lcvntr  |-  ( ph  ->  -.  R C U )

Proof of Theorem lcvntr
StepHypRef Expression
1 lcvnbtwn.s . . . 4  |-  S  =  ( LSubSp `  W )
2 lcvnbtwn.c . . . 4  |-  C  =  (  <oLL  `  W )
3 lcvnbtwn.w . . . 4  |-  ( ph  ->  W  e.  X )
4 lcvnbtwn.r . . . 4  |-  ( ph  ->  R  e.  S )
5 lcvnbtwn.t . . . 4  |-  ( ph  ->  T  e.  S )
6 lcvnbtwn.d . . . 4  |-  ( ph  ->  R C T )
71, 2, 3, 4, 5, 6lcvpss 29822 . . 3  |-  ( ph  ->  R  C.  T )
8 lcvnbtwn.u . . . 4  |-  ( ph  ->  U  e.  S )
9 lcvntr.p . . . 4  |-  ( ph  ->  T C U )
101, 2, 3, 5, 8, 9lcvpss 29822 . . 3  |-  ( ph  ->  T  C.  U )
117, 10jca 519 . 2  |-  ( ph  ->  ( R  C.  T  /\  T  C.  U ) )
123adantr 452 . . . 4  |-  ( (
ph  /\  R C U )  ->  W  e.  X )
134adantr 452 . . . 4  |-  ( (
ph  /\  R C U )  ->  R  e.  S )
148adantr 452 . . . 4  |-  ( (
ph  /\  R C U )  ->  U  e.  S )
155adantr 452 . . . 4  |-  ( (
ph  /\  R C U )  ->  T  e.  S )
16 simpr 448 . . . 4  |-  ( (
ph  /\  R C U )  ->  R C U )
171, 2, 12, 13, 14, 15, 16lcvnbtwn 29823 . . 3  |-  ( (
ph  /\  R C U )  ->  -.  ( R  C.  T  /\  T  C.  U ) )
1817ex 424 . 2  |-  ( ph  ->  ( R C U  ->  -.  ( R  C.  T  /\  T  C.  U ) ) )
1911, 18mt2d 111 1  |-  ( ph  ->  -.  R C U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C. wpss 3321   class class class wbr 4212   ` cfv 5454   LSubSpclss 16008    <oLL clcv 29816
This theorem is referenced by:  lsatcv0eq  29845
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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