Let $\mathcal{A}$ be a $\sigma$-algebra. Show that if $|\mathcal{A}| = \infty$, then $\mathcal{A}$ is uncountable.
We want to construct an infinite sequence of nonempty disjoint measurable sets. Then we use that to show that $\mathcal{A}$ contains uncountably many sets.
Let $A_1 \in \mathcal{A}$ which is neither $\emptyset$ nor $X$. Without loss of generality $A_1^\text{c}$ has an infinite number of measurable subsets.
My question is, why can we say the above bolded text? I imagine it ultimately follows from the assumption that $\mathcal{A}$ is infinite, but I'm not seeing the precise chain of logic. Could anybody help?