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Theorem elatcv0 29756
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 29754 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21baib 533 1 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2166   class class class wbr 4874   C cch 28342  0c0h 28348   ccv 28377  HAtomscat 28378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-at 29753
This theorem is referenced by: (None)
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