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Theorem psubatN 34059
 Description: A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
atpsub.a 𝐴 = (Atoms‘𝐾)
atpsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
psubatN ((𝐾𝐵𝑋𝑆𝑌𝑋) → 𝑌𝐴)

Proof of Theorem psubatN
StepHypRef Expression
1 atpsub.a . . . 4 𝐴 = (Atoms‘𝐾)
2 atpsub.s . . . 4 𝑆 = (PSubSp‘𝐾)
31, 2psubssat 34058 . . 3 ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)
43sseld 3567 . 2 ((𝐾𝐵𝑋𝑆) → (𝑌𝑋𝑌𝐴))
543impia 1253 1 ((𝐾𝐵𝑋𝑆𝑌𝑋) → 𝑌𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Atomscatm 33568  PSubSpcpsubsp 33800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-psubsp 33807 This theorem is referenced by: (None)
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