Let A, B, C be sets..

Prove that (AUB)∩ C is contained in AU(B∩ C)

I know how to show this using the venn diagram, and one professor I had told us just drawing the venn diagram is proof enough, but I think this professor wants a more formal proof.

My idea is to maybe let x belong to (AUB)∩ C and then this is like (A∩ C)U(B∩ C) and then A∩ C contains AU(B∩ C), but I feel like I am just running in circles.

Thanks for any help.