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Theorem atcv0 32089
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 32086 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21simprbi 496 1 (𝐴 ∈ HAtoms → 0 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5139   C cch 30676  0c0h 30682   ccv 30711  HAtomscat 30712
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 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-at 32085
This theorem is referenced by:  atcveq0  32095  atcv0eq  32126
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