There is the theorem:
Every orthonormal set $B$ in a Hilbert space $H$ is contained in a maximal orthonormal set in $H$.
There is the proof:
Let $\mathscr P$ be the class of all orthonormal sets in $H$ which contain the given set $B$. Partially order $\mathscr P$ by set inclusion.($\subset$)
Since $B$ $\in$ $\mathscr P$, $\mathscr P$ $\neq$ $\emptyset$.
Hence $\mathscr P$ contains a maximal totally ordered class $\Omega$.
Let $S$ be the union of all members of $\Omega$.
It is clear that $B$ $\subset$ $S$.
I don't understand how it's clear that $B$ $\subset$ $S$ ?
Any help would be appreciated.