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Theorem atcv0 30123
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 30120 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21simprbi 500 1 (𝐴 ∈ HAtoms → 0 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114   class class class wbr 5042   C cch 28710  0c0h 28716   ccv 28745  HAtomscat 28746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-un 3913  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-at 30119
This theorem is referenced by:  atcveq0  30129  atcv0eq  30160
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