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Theorem ela 29507
 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 4808 . 2 (𝑥 = 𝐴 → (0 𝑥 ↔ 0 𝐴))
2 df-at 29506 . 2 HAtoms = {𝑥C ∣ 0 𝑥}
31, 2elrab2 3507 1 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2139   class class class wbr 4804   Cℋ cch 28095  0ℋc0h 28101   ⋖ℋ ccv 28130  HAtomscat 28131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-at 29506 This theorem is referenced by:  elat2  29508  elatcv0  29509  atcv0  29510
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