How to write there exists exactly 1 x in a domain with property p without the unique quantifier? I need to write the statement 'there exists exactly one x in a domain such that p(x) is true'. This needs to be done without using the uniqueness quantifier $\exists!$. I've been staring at this problem for a while and I honestly haven't a clue how to approach this. 

Something like:
$$
\exists x \in X,\; (p(x) \wedge (\forall y \in X,\; p(y) \rightarrow y=x))
$$
That is, there is at least one element for which $p$ holds, and any element for which $p$ holds is the same.
