Theorem elatcv0 32243

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 32241	. 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 2109   class class class wbr 5102   C_ℋ cch 30831  0_ℋc0h 30837   ⋖_ℋ ccv 30866  HAtomscat 30867

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-at 32240

This theorem is referenced by: (None)