Proof of The Intersection of the Empty Set with Any Class Is Empty

By The Boolean Operations on Classes, Disjointness, and the Universal Class (i), $\emptyset\cap X=\{u:u\in\emptyset\wedge u\in X\}$, so by Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (i) a set $u$ is an element of $\emptyset\cap X$ if and only if

$$u\in\emptyset\quad\text{and}\quad u\in X.$$

An element $u$ of $\emptyset\cap X$ would therefore be an element of $\emptyset$; but $\emptyset$ has no element by The Empty Set, the Unordered Pair and the Singleton (i). Hence $\emptyset\cap X$ is a class with no element, and so it equals $\emptyset$ by Elements Are Sets, and the Empty Set and the Pair of Two Sets Exist and Are Unique (ii). This proves (i).

The same argument, with the roles of the two factors exchanged, gives $X\cap\emptyset=\emptyset$, which is (ii).