Theorem atcv0 32371

Description: An atom covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
atcv0	⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)

Proof of Theorem atcv0

Step	Hyp	Ref	Expression
1		ela 32368	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	simprbi 496	1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106   class class class wbr 5148   C_ℋ cch 30958  0_ℋc0h 30964   ⋖_ℋ ccv 30993  HAtomscat 30994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-at 32367

This theorem is referenced by:  atcveq0  32377  atcv0eq  32408