Uniformity iff Quasiuniformity has Symmetric Basis

Theorem
Let $S$ be a set.

Let $\UU$ be a quasiuniformity on $S$.

Then $\UU$ is a uniformity $\UU$ has a symmetric filter basis.

Proof
Let $\UU$ be a quasiuniformity on $S$ which has a symmetric filter basis $\BB$.

From the definition of filter basis, all the elements of $\UU$ can be formed from intersections of elements of $\BB$.

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

Now suppose $\UU$ is a uniformity.

If $\BB$ is a filter basis of $\UU$ then all the elements of $\BB$ are also elements of $\UU$.

Hence $\BB$ is a symmetric filter basis of $\UU$.