Prove NP-completeness for union of NP-complete language and language in P

Given disjoint languages $ X$ and $ Y$ , where $ X$ is NP-complete and $ Y\in P$ , how do I prove that $ X\cup Y$ is NP-complete?

My idea is to prove that $ (X\cup Y)\in NP$ and then prove that $ X\cup Y$ is NP-hard:

I can prove $ (X\cup Y)\in NP$ by saying that since both are in NP, they each have polytime verifiers. Thus we can have a polytime verifier that on any input will use these verifiers and accept if any accept, reject otherwise.

I am stuck on the part of proving NP-hardness. The idea of arbitrary languages is throwing me off; I am trying to reduce a NP-complete problem (SAT for example) to $ X\cup Y$ and using the fact that $ X$ has some method to reduce to it already but I am lost as for what to do with $ Y$ . I am thinking that given an input for SAT, I need to somehow change the input so that I can relate acceptance in $ X\cup Y$ to acceptance in SAT; same for rejection.

Any guidance would be appreciated; thanks!