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Theorem ela 23690
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 4157 . 2  |-  ( x  =  A  ->  ( 0H  <oH  x  <->  0H  <oH  A ) )
2 df-at 23689 . 2  |- HAtoms  =  {
x  e.  CH  |  0H  <oH  x }
31, 2elrab2 3037 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   class class class wbr 4153   CHcch 22280   0Hc0h 22286    <oH ccv 22315  HAtomscat 22316
This theorem is referenced by:  elat2  23691  elatcv0  23692  atcv0  23693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-at 23689
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