Theorem atcv0 32271

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 32268	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	simprbi 496	1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109   class class class wbr 5107   C_ℋ cch 30858  0_ℋc0h 30864   ⋖_ℋ ccv 30893  HAtomscat 30894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-at 32267

This theorem is referenced by:  atcveq0  32277  atcv0eq  32308