If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$

Question

Claim: If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$

$\coprod$ is the disjoint union of disjoint sets $A_n \subset A, \forall n \in \mathbb{N}$

Is there any more to this question other than countable union of countable set is countable, therefore $A_n$ cannot all be countabe if $A$ is uncountable?