Union of Power Sets

Theorem
The union of the power sets of two sets $$S$$ and $$T$$ is a subset of the power set of their union:

$$\mathcal{P} \left({S}\right) \cup \mathcal{P} \left({T}\right) \subseteq \mathcal{P} \left({S \cup T}\right)$$

Proof
$$ $$ $$ $$

Comment
Note that it is not always the case that $$\mathcal{P} \left({S}\right) \cup \mathcal{P} \left({T}\right) = \mathcal{P} \left({S \cup T}\right)$$.

Let $$S = \left\{{1, 2, 3}\right\}, T = \left\{{2, 3, 4}\right\}, X = \left\{{1, 2, 3, 4}\right\}$$.

$$ $$ $$

But note that $$X \not \subseteq S \land X \not \subseteq T$$.

Thus:

$$ $$ $$ $$ $$

So $$\mathcal{P} \left({S \cup T}\right) \ne \mathcal{P} \left({S}\right) \cup \mathcal{P} \left({T}\right)$$.