I am having problems resolving the following question:

Given some problem $X$. If there exists a polynomial time reduction from (for example) $\mbox{SAT}$ to $X$, $(\mbox{SAT} \leq_{p} X)$ and since we know that $\mbox{SAT}$ is $\mbox{NP-complete}$, to show that $X$ is $\mbox{NP-complete}$ is it necessary to show that $X\in \mbox{NP}$ via some third party algorithm?

If yes, then why?

Thank you