Let $A,B$ be sets. Prove $\forall A,B(\:\mathcal{P}(A)\cup\mathcal{P}(B)\subset\mathcal{P}(A\cup B)\:)$
Attempt:
$a\in \mathcal{P}(A)\cup\mathcal{P}(B)\iff a\in\mathcal{P}(A)\vee a\in \mathcal{P}(B) \iff a\subset A\vee a\subset B\iff a\subset A\cup B\iff a\in \mathcal{P}(A\cup B)$
Comment:
This proof is wrong as it turns out that equality is impossible, though I don't understand why the presented logic isn't logical.
On
Consider $A=\{1,2,3\}$ and $B=\{4,5\}$. So $\{3,4\}\subset A\cup B$ but this does not imply that $\{3,4\}\subset A\vee \{3,4\}\subset B$. So, your third equivalence is not true in general.
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One way to investigate dubious proofs is to take a concrete example and spot where the reasoning fails. In this case, observe that $A = \{0\}, B = \{1\}$ is a counterexample to the original statement if $a = \{0,1\}$ since $a \in \mathcal{P}(A \cup B)$ but $a \notin \mathcal{P}(A) \cup \mathcal{P}(B)$. Can you see where the proof fails?
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Suppose $X\subset Y$. Then prove that $\mathcal{P}(X)\subset\mathcal{P}(Y)$.
In particular, $\mathcal{P}(A)\subset\mathcal{P}(A\cup B)$ and $\mathcal{P}(B)\subset\mathcal{P}(A\cup B)$.
Therefore $\mathcal{P}(A)\cup\mathcal{P}(B)\subset\mathcal{P}(A\cup B)$.
You have a wrong proof because you're using a wrong double implication. You can have $a\subset A\cup B$, but $a\not\subset A$ and $a\not\subset B$.
The correct implications: $$ a\in \mathcal{P}(A)\cup\mathcal{P}(B)\iff a\in\mathcal{P}(A)\vee a\in \mathcal{P}(B) \iff a\subset A\vee a\subset B\implies a\subset A\cup B\iff a\in \mathcal{P}(A\cup B). $$ The iffs are consequence on the definition of $\cup$ and $\mathcal{P}$. The implication $$ a\subset A\cup B\implies a\subset A\vee a\subset B $$ is false. Can you see why? Hint: $a$ must be "divided" between $A$ and $B$.