Trick Proof: relation of equivalence

Question

What is wrong in the following reasoning:

Every symmetric and transitive relation is a relation of equivalence

Proof:

$x \sim y \Rightarrow y\sim x$ - becuase of symmetry

$x \sim y \wedge y\sim x \Rightarrow x\sim x$ - because of transitivity

Therefore the relation is reflexive - so it is a relation of equivalence

Answer 1

This argument assumes that $x$ is related to some $y$, which may not be the case.

Answer 2

I would elaborate the last answer. The set of things that particular x can be relative to(and due to symmetry those are relative to x) can be Empty set. If you write your argument fully mathematically - symmetry: For any(means if there is any) x ~ y => y ~ x. As a result, you may not have anything to input in this chain of logic to establish the output, which is otherwise correct.