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Theorem pclssidN 37023
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 4885 . 2 𝑋 {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦}
2 pclss.a . . 3 𝐴 = (Atoms‘𝐾)
3 eqid 2819 . . 3 (PSubSp‘𝐾) = (PSubSp‘𝐾)
4 pclss.c . . 3 𝑈 = (PCl‘𝐾)
52, 3, 4pclvalN 37018 . 2 ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦 ∈ (PSubSp‘𝐾) ∣ 𝑋𝑦})
61, 5sseqtrrid 4018 1 ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  {crab 3140  wss 3934   cint 4867  cfv 6348  Atomscatm 36391  PSubSpcpsubsp 36624  PClcpclN 37015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-psubsp 36631  df-pclN 37016
This theorem is referenced by:  pclunN  37026  pcl0bN  37051  pclfinclN  37078
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