Theorem atcv0 29777

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 29774	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	simprbi 492	1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107   class class class wbr 4888   C_ℋ cch 28362  0_ℋc0h 28368   ⋖_ℋ ccv 28397  HAtomscat 28398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-at 29773

This theorem is referenced by:  atcveq0  29783  atcv0eq  29814