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Theorem ela 23795
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 4176 . 2  |-  ( x  =  A  ->  ( 0H  <oH  x  <->  0H  <oH  A ) )
2 df-at 23794 . 2  |- HAtoms  =  {
x  e.  CH  |  0H  <oH  x }
31, 2elrab2 3054 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   class class class wbr 4172   CHcch 22385   0Hc0h 22391    <oH ccv 22420  HAtomscat 22421
This theorem is referenced by:  elat2  23796  elatcv0  23797  atcv0  23798
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-at 23794
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