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Theorem lssel 20935
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 ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)

Proof of Theorem lssel
StepHypRef Expression
1 lssss.v . . 3 𝑉 = (Base‘𝑊)
2 lssss.s . . 3 𝑆 = (LSubSp‘𝑊)
31, 2lssss 20934 . 2 (𝑈𝑆𝑈𝑉)
43sselda 3983 1 ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cfv 6561  Basecbs 17247  LSubSpclss 20929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-lss 20930
This theorem is referenced by:  lssvacl  20941  lssvsubcl  20942  lssvancl1  20943  lssvancl2  20944  lss0cl  20945  lssvscl  20953  lssvnegcl  20954  ellspsn6  20992  ellspsn5  20994  lssats2  20998  lsmcl  21082  lsmelval2  21084  lsmcv  21143  ocvin  21692  lsatel  39006  lsmsat  39009  lssatomic  39012  lssats  39013  lsat0cv  39034  lshpkrlem1  39111  lshpkrlem5  39115  lshpkr  39118  dihjat1lem  41430  dochsatshpb  41454  lcfrvalsnN  41543  lcfrlem4  41547  lcfrlem6  41549  lcfrlem16  41560  lcfrlem29  41573  lcfrlem35  41579  mapdval4N  41634  mapdpglem2a  41676  mapdpglem23  41696
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