Uniformity iff Quasiuniformity has Symmetric Basis

Theorem
Let $S$ be a set.

Let $\mathcal U$ be a quasiuniformity on $S$.

Then $\mathcal U$ is a uniformity $\mathcal U$ has a symmetric filter basis.

Proof
Let $\mathcal U$ be a quasiuniformity on $S$ which has a symmetric filter basis $\mathcal B$.

From the definition of filter basis, all the elements of $\mathcal U$ can be formed from intersections of elements of $\mathcal B$.

But from Intersection of Symmetric Relations is Symmetric, it follows that all elements of $\mathcal U$ are symmetric.

Now suppose $\mathcal U$ is a uniformity.

If $\mathcal B$ is a filter basis of $\mathcal U$ then all the elements of $\mathcal B$ are also elements of $\mathcal U$.

Hence $\mathcal B$ is a symmetric filter basis of $\mathcal U$.