Theorem atcv0 32603

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 32600	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	simprbi 502	1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2145   class class class wbr 5105   C_ℋ cch 31190  0_ℋc0h 31196   ⋖_ℋ ccv 31225  HAtomscat 31226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-at 32599

This theorem is referenced by:  atcveq0  32609  atcv0eq  32640