Orthocomplement of Subset of Orthocomplement is Superset

Theorem
Let $H$ be a Hilbert space.

Let $A, B \subseteq H$ be subsets of $H$ such that $B \subseteq A^\perp$, where $A^\perp$ is the orthocomplement of $A$.

Then $A \subseteq B^\perp$.

Proof
Let $B \subseteq A^\perp$.

Then by Orthocomplement Reverses Subset:


 * $A^{\perp\perp} \subseteq B^\perp$

By Double Orthocomplement is Closed Linear Span and the definition of closed linear span:


 * $A \subseteq A^{\perp\perp}$

Hence, by Subset Relation is Transitive:


 * $A \subseteq B^\perp$