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Theorem ela 22921
Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
ela  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )

Proof of Theorem ela
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq2 4029 . 2  |-  ( x  =  A  ->  ( 0H  <oH  x  <->  0H  <oH  A ) )
2 df-at 22920 . 2  |- HAtoms  =  {
x  e.  CH  |  0H  <oH  x }
31, 2elrab2 2927 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1686   class class class wbr 4025   CHcch 21511   0Hc0h 21517    <oH ccv 21546  HAtomscat 21547
This theorem is referenced by:  elat2  22922  elatcv0  22923  atcv0  22924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-at 22920
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