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Theorem elatcv0 32169
Description: A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
elatcv0 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))

Proof of Theorem elatcv0
StepHypRef Expression
1 ela 32167 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21baib 534 1 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098   class class class wbr 5150   C cch 30757  0c0h 30763   ccv 30792  HAtomscat 30793
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 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-at 32166
This theorem is referenced by: (None)
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