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.

Prove (A union B) intersect C is a subset of A union (B intersect C)
$\displaystyle \begin{gathered} x \in \left[ {\left( {A \cup B} \right) \cap C} \right] \hfill \\ x \in (A \cup B) \wedge x \in C \hfill \\ \left[ {x \in A \vee x \in B} \right] \wedge x \in C \hfill \\ \left[ {\left( {x \in A \wedge x \in C} \right) \vee \left( {x \in B \wedge x \in C} \right)} \right] \hfill \\ \end{gathered}$

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a union (b intersect c) is a subset (a union b) intersect c

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