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Metamath Proof Explorer

Theorem hatomici

Description: The Hilbert lattice is atomic, i.e. any nonzero element is greater than or equal to some atom. Remark in Kalmbach p. 140. (Contributed by NM, 22-Jul-2001) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	hatomic.1	$⊢ A \in C_{ℋ}$
	Assertion	hatomici	$⊢ A \neq 0_{ℋ} \to \exists x \in HAtoms x \subseteq A$

Proof

Step	Hyp	Ref	Expression
1		hatomic.1	$⊢ A \in C_{ℋ}$
2	1	chshii	$⊢ A \in S_{ℋ}$
3	2	shatomici	$⊢ A \neq 0_{ℋ} \to \exists x \in HAtoms x \subseteq A$