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Theorem atelch 31628
Description: An atom is a Hilbert lattice element. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
atelch (𝐴 ∈ HAtoms → 𝐴C )

Proof of Theorem atelch
StepHypRef Expression
1 atssch 31627 . 2 HAtoms ⊆ C
21sseli 3979 1 (𝐴 ∈ HAtoms → 𝐴C )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   C cch 30213  HAtomscat 30249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-at 31622
This theorem is referenced by:  atsseq  31631  atcveq0  31632  chcv1  31639  chcv2  31640  hatomistici  31646  chrelati  31648  chrelat2i  31649  cvati  31650  cvexchlem  31652  cvp  31659  atnemeq0  31661  atcv0eq  31663  atcv1  31664  atexch  31665  atomli  31666  atoml2i  31667  atordi  31668  atcvatlem  31669  atcvati  31670  atcvat2i  31671  chirredlem1  31674  chirredlem2  31675  chirredlem3  31676  chirredlem4  31677  chirredi  31678  atcvat3i  31680  atcvat4i  31681  atdmd  31682  atmd  31683  atmd2  31684  atabsi  31685  mdsymlem2  31688  mdsymlem3  31689  mdsymlem5  31691  mdsymlem8  31694  atdmd2  31698  sumdmdi  31704  dmdbr4ati  31705  dmdbr5ati  31706  dmdbr6ati  31707
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