Adding an Element to, Subsets of, and Unions and Cartesian Products of Finite Sets Are Finite

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let finite sets be as in Finite Sets §finite, and let $A$ and $B$ be sets.

¶ If $A$ is finite, then $A\cup\{x\}$ is finite for every set $x$.

¶ If $A$ is finite, every subset of $A$ is finite.

¶ If $A$ and $B$ are finite, then $A\cup B$ and $A\times B$ are finite.