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Theorem atcv0 31582
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 31579 . 2 (𝐴 ∈ HAtoms ↔ (𝐴C ∧ 0 𝐴))
21simprbi 497 1 (𝐴 ∈ HAtoms → 0 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 5147   C cch 30169  0c0h 30175   ccv 30204  HAtomscat 30205
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-at 31578
This theorem is referenced by:  atcveq0  31588  atcv0eq  31619
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