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Theorem atelch 29533
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 29532 . 2 HAtoms ⊆ C
21sseli 3740 1 (𝐴 ∈ HAtoms → 𝐴C )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139   C cch 28116  HAtomscat 28152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-in 3722  df-ss 3729  df-at 29527
This theorem is referenced by:  atsseq  29536  atcveq0  29537  chcv1  29544  chcv2  29545  hatomistici  29551  chrelati  29553  chrelat2i  29554  cvati  29555  cvexchlem  29557  cvp  29564  atnemeq0  29566  atcv0eq  29568  atcv1  29569  atexch  29570  atomli  29571  atoml2i  29572  atordi  29573  atcvatlem  29574  atcvati  29575  atcvat2i  29576  chirredlem1  29579  chirredlem2  29580  chirredlem3  29581  chirredlem4  29582  chirredi  29583  atcvat3i  29585  atcvat4i  29586  atdmd  29587  atmd  29588  atmd2  29589  atabsi  29590  mdsymlem2  29593  mdsymlem3  29594  mdsymlem5  29596  mdsymlem8  29599  atdmd2  29603  sumdmdi  29609  dmdbr4ati  29610  dmdbr5ati  29611  dmdbr6ati  29612
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