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

Let $x$ and $y$ be sets and $Y$ a class.

¶ Every subclass of $x$ is a set. In particular the intersection $x\cap Y$ and the difference $x\setminus Y$ are sets.

¶ There is exactly one set $z$ such that, for every set $u$, $u\in z$ if and only if there is a set $v$ with $u\in v$ and $v\in x$.

¶ There is exactly one set $z$ such that, for every set $u$, $u\in z$ if and only if $u$ is a subset of $x$.

¶ The union $x\cup y$ is a set.

¶ The universal class $V$ is a proper class.