Infimum of Intersection of Upper Closures equals Join Operands

Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice.

Let $x, y \in S$.

Then $\inf\left({x^\succeq \cap y^\succeq}\right) = x \vee y$

Proof
By Intersection of Upper Closures is Upper Closure of Join Operands:
 * $x^\succeq \cap y^\succeq = \left({x \vee y}\right)^\succeq$

Thus by Infimum of Upper Closure of Element:
 * $\inf\left({x^\succeq \cap y^\succeq}\right) = x \vee y$