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Theorem atcv0 28585
Description: An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
atcv0 (𝐴 ∈ HAtoms → 0 𝐴)

Proof of Theorem atcv0
StepHypRef Expression
1 ela 28582 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21simprbi 479 1 (𝐴 ∈ HAtoms → 0 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977   class class class wbr 4583   C cch 27170  0c0h 27176   ccv 27205  HAtomscat 27206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-at 28581
This theorem is referenced by:  atcveq0  28591  atcv0eq  28622
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