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Theorem ela 30097
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 5042 . 2 (𝑥 = 𝐴 → (0 𝑥 ↔ 0 𝐴))
2 df-at 30096 . 2 HAtoms = {𝑥C ∣ 0 𝑥}
31, 2elrab2 3659 1 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wcel 2114   class class class wbr 5038   C cch 28687  0c0h 28693   ccv 28722  HAtomscat 28723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-at 30096
This theorem is referenced by:  elat2  30098  elatcv0  30099  atcv0  30100
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