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Theorem lssel 15711
Description: A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lssss.v  |-  V  =  ( Base `  W
)
lssss.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssel  |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )

Proof of Theorem lssel
StepHypRef Expression
1 lssss.v . . 3  |-  V  =  ( Base `  W
)
2 lssss.s . . 3  |-  S  =  ( LSubSp `  W )
31, 2lssss 15710 . 2  |-  ( U  e.  S  ->  U  C_  V )
43sselda 3193 1  |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271   Basecbs 13164   LSubSpclss 15705
This theorem is referenced by:  lssvsubcl  15717  lssvancl1  15718  lssvancl2  15719  lss0cl  15720  lssvacl  15727  lssvscl  15728  lssvnegcl  15729  lspsnel6  15767  lspsnel5a  15769  lssats2  15773  lsmcl  15852  lsmelval2  15854  lsmcv  15910  ocvin  16590  lsatel  29817  lsmsat  29820  lssatomic  29823  lssats  29824  lsat0cv  29845  lshpkrlem1  29922  lshpkrlem5  29926  lshpkr  29929  dihjat1lem  32240  dochsatshpb  32264  lcfrvalsnN  32353  lcfrlem4  32357  lcfrlem6  32359  lcfrlem16  32370  lcfrlem29  32383  lcfrlem35  32389  mapdval4N  32444  mapdpglem2a  32486  mapdpglem23  32506
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
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-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-ov 5877  df-lss 15706
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