Condition for Set Union Equivalent to Associated Cardinal Number

Theorem
Let $S$ and $T$ be sets.

Let $|S|$ denote the cardinal number of $S$.

Then:


 * $S \cup T \sim | S \cup T | \iff S \sim | S | \land T \sim | T |$

Necessary Condition
If $S \cup T \sim | S \cup T |$, then there is a mapping $f$ such that:


 * $f : S \cup T \to | S \cup T |$ is a bijection.

Since $f$ is a bijection, it follows that:


 * $S$ is equivalent to the image of $S$ under $f$.

This, in turn, is a subset of the ordinal $| S \cup T |$.

$| S \cup T |$ is an ordinal by Cardinal Number is Ordinal Number.

By Condition for Set Equivalent to Cardinal Number, it follows that $S \sim |S|$.

Similarly, $T \sim |T|$.

Sufficient Condition
Suppose $f : S \to |S|$ is a bijection and $g : T \to |T|$ is a bijection.

Define the function $F$ to be:


 * $F\left({ x }\right) = f\left({x}\right)$ if $x \in S$.


 * $F\left({ x }\right) = |S| + g\left({x}\right)$ if $x \notin S \land x \in T$.

It follows that $F : S \cup T \to |S| + |T|$ is an injection.

Therefore, $S \cup T$ is equivalent to some subset of the ordinal $|S| + |T|$.

By Condition for Set Equivalent to Cardinal Number, it follows that $S \cup T \sim |S \cup T|$.