Prove (A union B) intersect C is a subset of A union (B intersect C)

x is an element of (A union B) intersect C. Thus, x is an element of A intersect B or an element of B intersect C or an element of A intersect B intersect C. If x is an element of A intersect B then x is an element of A union (B intersect C). If x is an element of B intersect C then x is an element of A union (B intersect C). If x is an element of A intersect B intersect C then x has to be an element of A union (B intersect C). Therefore, x will always be an element of A union (B intersect C).

I feel like the logic is correct but i may have skipped some steps. Any comments would be great.