Theorem ela

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.)

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

Proof

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