Theorem atcv0 32977

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 32974	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	simprbi 503	1 ⊢ (𝐴 ∈ HAtoms → 0ℋ ⋖ℋ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   class class class wbr 5106   C_ℋ cch 31564  0_ℋc0h 31570   ⋖_ℋ ccv 31599  HAtomscat 31600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-at 32973

This theorem is used by:  atcveq0  32983  atcv0eq  33014