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Metamath Proof Explorer

Theorem chrelat3i

Description: A consequence of the relative atomicity of Hilbert space: the ordering of Hilbert lattice elements is completely determined by the atoms they majorize. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chrelat3.1	$⊢ A \in C_{ℋ}$
	chrelat3.2	$⊢ B \in C_{ℋ}$
Assertion	chrelat3i	$⊢ A \subseteq B \leftrightarrow \forall x \in HAtoms (x \subseteq A \to x \subseteq B)$

Proof

Step	Hyp	Ref	Expression
1		chrelat3.1	$⊢ A \in C_{ℋ}$
2		chrelat3.2	$⊢ B \in C_{ℋ}$
3		chrelat3	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to (A \subseteq B \leftrightarrow \forall x \in HAtoms (x \subseteq A \to x \subseteq B))$
4	1 2 3	mp2an	$⊢ A \subseteq B \leftrightarrow \forall x \in HAtoms (x \subseteq A \to x \subseteq B)$