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	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chrelat3.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chrelat3i	$$⊢A\subseteq B\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq A\to x\subseteq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		chrelat3.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chrelat3.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3		chrelat3	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in {\mathbf{C}}_{\mathscr{H}}\right)\to \left(A\subseteq B\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq A\to x\subseteq B\right)\right)$$
4	1 2 3	mp2an	$$⊢A\subseteq B\leftrightarrow \forall x\in \mathrm{HAtoms}\left(x\subseteq A\to x\subseteq B\right)$$