If $|A| = |A_1| = |A_2|$ holds for infinite sets $A, A_1$ and $A_2$ where $A_1 \cap A_2 = \emptyset$, is it true that $|A| = |A_1 \cup A_2|?$

Question

For some reason, I think this is true. I suppose it is clear why $|A| \leq |A_1 \cup A_2|$. I wonder if there's a way to show the existence of an injection $\phi: A_1 \cup A_2 \to A$.

I would appreciate any help. Thanks in advance.