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Theorem ela 22919
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 4027 . 2  |-  ( x  =  A  ->  ( 0H  <oH  x  <->  0H  <oH  A ) )
2 df-at 22918 . 2  |- HAtoms  =  {
x  e.  CH  |  0H  <oH  x }
31, 2elrab2 2925 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   class class class wbr 4023   CHcch 21509   0Hc0h 21515    <oH ccv 21544  HAtomscat 21545
This theorem is referenced by:  elat2  22920  elatcv0  22921  atcv0  22922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-at 22918
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