Theorem elatcv0 32365

Description: A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
elatcv0	⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴))

Proof of Theorem elatcv0

Step	Hyp	Ref	Expression
1		ela 32363	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	baib 535	1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113   class class class wbr 5096   C_ℋ cch 30953  0_ℋc0h 30959   ⋖_ℋ ccv 30988  HAtomscat 30989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-at 32362

This theorem is referenced by: (None)