Why the opposite direction of proving $\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$ is wrong

The following proof is given to show that $$\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$$ ($$\mathcal P$$ is the power set):

Let $$X \in (\mathcal{P}(A) \cup \mathcal{P}(B))$$, by definition of union: $$X \in \mathcal{P}(A) \lor X \in \mathcal{P}(B)$$ $$\implies$$By definition of power set $$X \subseteq A \lor X \subseteq B$$ $$\implies$$By union transitivity$$X \subseteq A\cup B$$ $$\implies$$By definition of power set $$X \in \mathcal{P}(A \cup B)$$ Then $$\mathcal{P}(A) \cup \mathcal{P}(B) \subseteq \mathcal{P}(A \cup B) \blacksquare$$

Now, I thought I could apply the opposite logic to show that $$\mathcal P(A) \cup \mathcal P(B) \supseteq \mathcal P(A\cup B)$$, and thus prove that $$\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$$:

Let $$X \in \mathcal{P}(A \cup B)$$, By definition of power set: $$X \subseteq A\cup B$$ $$\implies$$ By union transitivity: $$X \subseteq A \lor X \subseteq B$$ $$\implies$$ By definition of powerset: $$X \in \mathcal{P}(A) \lor X \in \mathcal{P}(B)$$ $$\implies$$ by definition of union: $$X \in (\mathcal{P}(A) \cup \mathcal{P}(B))$$ Then $$\mathcal {P}(A \cup B) \subseteq \mathcal{P}(A) \cup \mathcal{P}(B) \blacksquare$$

Multiple posts here prove that for any sets $$A$$ or $$B$$, if $$\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$$ then either $$A \subseteq B$$ or $$B \subseteq A$$, so clearly my opposite direction proof can't be true.

What I'd like to understand is which step is based on a false assumtion/implication?

3 Answers

$$X \subseteq A \cup B$$ doesn't imply $$X \subseteq A$$ or $$X \subseteq B$$.

For example let $$A=\{1\}$$ and $$B=\{2\}$$ and $$X = A \cup B$$.

$$X$$ is neither subset of $$A$$ nor subset of $$B$$.

Your "union transitivity" step is wrong. That is, if you have $$X \subset A \cup B$$, then you don't necessarily have $$X \subset A$$ or $$A \subset B$$. For example, let $$A =\{1\},B=\{2\},X=\{1,2\}$$.

I have lot of books in my shelf: Some of them I bought and the rest are borrowed from others and never returned.

A general subset of books from my shelf need not consist exclusively of borrowed books, or of own books.