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Theorem ela 28582
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 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))

Proof of Theorem ela
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq2 4587 . 2 (𝑥 = 𝐴 → (0 𝑥 ↔ 0 𝐴))
2 df-at 28581 . 2 HAtoms = {𝑥C ∣ 0 𝑥}
31, 2elrab2 3333 1 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wcel 1977   class class class wbr 4583   C cch 27170  0c0h 27176   ccv 27205  HAtomscat 27206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-at 28581
This theorem is referenced by:  elat2  28583  elatcv0  28584  atcv0  28585
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