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Theorem ela 23799
 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 HAtoms

Proof of Theorem ela
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq2 4180 . 2
2 df-at 23798 . 2 HAtoms
31, 2elrab2 3058 1 HAtoms
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wcel 1721   class class class wbr 4176  cch 22389  c0h 22395   ccv 22424  HAtomscat 22425 This theorem is referenced by:  elat2  23800  elatcv0  23801  atcv0  23802 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 2389 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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-at 23798
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