Let $(A_\gamma)_{\gamma<\alpha}$ and $(B_\gamma)_{\gamma<\alpha}$ be two sequence of sets. They satisfy $|A_\gamma|=|B_\gamma|$ for all $\gamma<\alpha$, where $|\cdot|$ denotes cardinality. Also, whenever $\beta<\gamma$, we have $A_\beta\subseteq A_\gamma$ and $B_\beta\subseteq B_\gamma$.
My question is: is it always true that $$ \left|\bigcup_{\gamma<\alpha}A_\gamma\right|=\left|\bigcup_{\gamma<\alpha}B_\gamma\right|? $$ I am struggling quite a bit when trying to prove this, but I could not find any counterexamples either. This is not given in my book, and I come up with this myself, so I really do not know where to start.
