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Theorem elatcv0 30103
 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 30101 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21baib 538 1 (𝐴C → (𝐴 ∈ HAtoms ↔ 0 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2114   class class class wbr 5042   Cℋ cch 28691  0ℋc0h 28697   ⋖ℋ ccv 28726  HAtomscat 28727 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 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-at 30100 This theorem is referenced by: (None)
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