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Theorem ela 22749
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
StepHypRef Expression
1 breq2 3924 . 2  |-  ( x  =  A  ->  ( 0H  <oH  x  <->  0H  <oH  A ) )
2 df-at 22748 . 2  |- HAtoms  =  {
x  e.  CH  |  0H  <oH  x }
31, 2elrab2 2862 1  |-  ( A  e. HAtoms 
<->  ( A  e.  CH  /\  0H  <oH  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1621   class class class wbr 3920   CHcch 21339   0Hc0h 21345    <oH ccv 21374  HAtomscat 21375
This theorem is referenced by:  elat2  22750  elatcv0  22751  atcv0  22752
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-at 22748
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