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Prove $(A\cup B)\setminus C\subseteq A\cup (B\setminus C)$

Let $x\in (A\cup B)\setminus C$. Then $x\in (A\setminus C)\cup (B\setminus C)$. Because $A\setminus C\subseteq A$, we have the final solution $x\in A\cup(B\setminus C)$.

I feel like this is correct but I would like an opinion of yours.

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    Looks fine to me ... Draw a Venn diagram – 2017-02-26
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    I agree it looks fine but i disagree that drawing a venn diagram should be your next action. Venn diagrams are often a crutch which can only help for smaller problems that prevent people from getting the necessary practice with formally written proofs that they need. Once you get into four or more sets in the problem statement, it becomes incredibly difficult or impossible to draw an appropriate venn diagram and even harder to manipulate it correctly. – 2017-02-26

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Perhaps with a few words it is clearer:

$$x\in (A\cup B)\setminus C\implies \left(x\in A\;\;\text{or}\;\;x\in B\right)\;\;\text{and}\;\;x\notin C$$

Then

$$(1)\;\text{ If}\;\;x\in A\;\;\text{then}\;\;x\in A\cup(B\setminus C)\;,\;\;(2)\;\text{and if}\;\;x\notin A\;\text{then}\;\;x\in B\;\;\text{and}\;\;x\notin C\implies$$

$$x\in B\setminus C\implies x\in A\cup(B\setminus C)$$