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Theorem ela 32061
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 5142 . 2 (𝑥 = 𝐴 → (0 𝑥 ↔ 0 𝐴))
2 df-at 32060 . 2 HAtoms = {𝑥C ∣ 0 𝑥}
31, 2elrab2 3678 1 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2098   class class class wbr 5138   C cch 30651  0c0h 30657   ccv 30686  HAtomscat 30687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-at 32060
This theorem is referenced by:  elat2  32062  elatcv0  32063  atcv0  32064
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