\begin{aligned}
 \bigcup_{x\in X} A_x
 &= \bigcup_{x=x_0} A_x \cup \bigcup_{x\in X\backslash\{x_0\}} A_x \\
 &= \{x_0\}\cup (A\backslash X) \cup (X\backslash\{x_0\}) \\
 &= (A\backslash X)\cup X \\
 &= A
\end{aligned}