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Theorem ela 32358
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 5147 . 2 (𝑥 = 𝐴 → (0 𝑥 ↔ 0 𝐴))
2 df-at 32357 . 2 HAtoms = {𝑥C ∣ 0 𝑥}
31, 2elrab2 3695 1 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108   class class class wbr 5143   C cch 30948  0c0h 30954   ccv 30983  HAtomscat 30984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-at 32357
This theorem is referenced by:  elat2  32359  elatcv0  32360  atcv0  32361
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