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Theorem pclssidN 39877
Description: A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclss.a 𝐴 = (Atoms‘𝐾)
pclss.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclssidN ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))

Proof of Theorem pclssidN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4970 . 2 𝑋 {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦}
2 pclss.a . . 3 𝐴 = (Atoms‘𝐾)
3 eqid 2734 . . 3 (PSubSp‘𝐾) = (PSubSp‘𝐾)
4 pclss.c . . 3 𝑈 = (PCl‘𝐾)
52, 3, 4pclvalN 39872 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
61, 5sseqtrrid 4048 1 ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {crab 3432  wss 3962   cint 4950  cfv 6562  Atomscatm 39244  PSubSpcpsubsp 39478  PClcpclN 39869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-psubsp 39485  df-pclN 39870
This theorem is referenced by:  pclunN  39880  pcl0bN  39905  pclfinclN  39932
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