trouble with proof in elementary set theory regarding powersets 
Let $X$ be an arbitrary set and $\mathcal{P}(X)$ be the power set of $X$. Prove that for every $A,B \in \mathcal{P}(X)$ we have $A\cup B \in \mathcal{P}(X)$. 

By the definition of the power set we know that $A,B \subseteq X$. Let's use the definition of subset:


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*$A, B \subseteq X \iff (\forall \space x \in A, y \in B \implies x, y \in X)$


We also have per definition that $\forall A,B \subseteq X$:


*

*$A \cup B = \{ x \in X | \space x \in A \lor x \in B\}$


Let  $z \in A \cup B$. Per definition, we know that  $\space z \in A \lor z \in B$. Since we have $(\forall \space x \in A, y \in B \implies x, y \in X)$, we also know that $z \in X$.  
Here I'm stuck. I don't know how to continue with this proof. Intuitively the result is logical but I don't know how to make something useful out of $z$. 

Don't forget what you're trying to prove.  You're trying to prove that $A\cup B\in\mathcal{P}(X)$.  By definition of $\mathcal{P}(X)$, this just means that $A\cup B\subseteq X$.  And by definition of $\subseteq$, this means that for any $z\in A\cup B$, $z\in X$.
So what you need to prove is that for any $z\in A\cup B$, $z\in X$.  But that's exactly what you did prove!  So you're done.
