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Theorem lcvpss 29836
Description: The covers relation implies proper subset. (cvpss 22881 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s  |-  S  =  ( LSubSp `  W )
lcvfbr.c  |-  C  =  (  <oLL  `  W )
lcvfbr.w  |-  ( ph  ->  W  e.  X )
lcvfbr.t  |-  ( ph  ->  T  e.  S )
lcvfbr.u  |-  ( ph  ->  U  e.  S )
lcvpss.d  |-  ( ph  ->  T C U )
Assertion
Ref Expression
lcvpss  |-  ( ph  ->  T  C.  U )

Proof of Theorem lcvpss
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lcvpss.d . . 3  |-  ( ph  ->  T C U )
2 lcvfbr.s . . . 4  |-  S  =  ( LSubSp `  W )
3 lcvfbr.c . . . 4  |-  C  =  (  <oLL  `  W )
4 lcvfbr.w . . . 4  |-  ( ph  ->  W  e.  X )
5 lcvfbr.t . . . 4  |-  ( ph  ->  T  e.  S )
6 lcvfbr.u . . . 4  |-  ( ph  ->  U  e.  S )
72, 3, 4, 5, 6lcvbr 29833 . . 3  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
81, 7mpbid 201 . 2  |-  ( ph  ->  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) )
98simpld 445 1  |-  ( ph  ->  T  C.  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C. wpss 3166   class class class wbr 4039   ` cfv 5271   LSubSpclss 15705    <oLL clcv 29830
This theorem is referenced by:  lcvntr  29838  lcvat  29842  lsatcveq0  29844  lsat0cv  29845  lcvexchlem4  29849  lcvexchlem5  29850  lcv1  29853  lsatexch  29855  lsatcvat2  29863  islshpcv  29865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lcv 29831
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