Theorem ela 32702

Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
ela	⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))

Proof of Theorem ela

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5112	. 2 ⊢ (𝑥 = 𝐴 → (0ℋ ⋖ℋ 𝑥 ↔ 0ℋ ⋖ℋ 𝐴))
2		df-at 32701	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
3	1, 2	elrab2 3653	1 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∈ wcel 2142   class class class wbr 5108   C_ℋ cch 31292  0_ℋc0h 31298   ⋖_ℋ ccv 31327  HAtomscat 31328

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-at 32701

This theorem is used by:  elat2  32703  elatcv0  32704  atcv0  32705