The Union Set and the Power Set of a Set

Let $x$ be a set.

¶ The union $\bigcup x$ of $x$ is the unique set whose elements are exactly the sets $u$ for which there is a set $v$ with $u\in v$ and $v\in x$, given by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §union.

¶ The power set $\mathcal{P}(x)$ of $x$ is the unique set whose elements are exactly the subsets of $x$, given by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §power.

Both are defined set symbols.