I'm trying to proof the following statement that my intuition tells me is true:
Let $A$ and $I$ be sets and $(A_i)_{i\in I}$ a family of sets with $|A_i|<|A|$ for all $i\in I$ and
$$A=\bigcup_{i\in I}A_i.$$ Then it holds that $|I|\ge cf(|A|)$.
Here are my thoughts: I know the statement is true if we require $|A_i|<cf|A|$ instead since in this case I can follow a known proof for regular cardinals. This proof is by contradiction and assumes $|I|< cf(|A|)$. Since no $A_i$ is cofinal in $A$ we can find a set $S=${$a_i\mid i\in I$} with $x<a_i$ for all $i\in I$ and $x\in A_i$. By assumption $S$ can't be cofinal in $A$ as well and we can find an $a\in A$ with $x<a$ for all $x\in A$ including $x=a$ which is a contradicton. My idea was to somehow stich together all $A_i$'s along a fixed well-order on $I$. Something like finding a ordinal $B$ bijective to $A$ that can be written as
$$B=\bigcup_{i\in I}B_i$$
but this time with $x<y$ for all $x\in B_i$ and $y\in B_j$ if $i<j$ holds. That way we could argure as before since again no $B_i$ should be cofinal in $B$. But I know there are sill some challanges (mainly finidng this ordinal and the case if there's a maximal $i\in I$ for which $A_i$ may not be cofinal as desired).
I hope my thoughts are understandable and wonder if my approach is leading anywhere. Also any hint in another direction would be much appreciated.
First, we know that either $\sup \{ \vert A_i \vert \mid i \in I \}= \vert A \vert$ or $\vert I \vert \geq \vert A \vert$. That's because if neither of those is true, then by the rules of cardinal arithmetic, we would have that $\vert A \vert = \left \vert \bigcup A_i \right \vert \leq \max \{ \sup \{ \vert A_i \vert \mid i \in I \}, \vert I \vert \} \lt \vert A \vert$, which is a contradiction.
But now, if $\vert I \vert \lt \vert A \vert$, we know that $(\vert A_i \vert \mid i \in I)$ is cofinal in $\vert A \vert$, so $\vert I \vert \geq \operatorname{cof} \vert A \vert $. Of course, if $\vert I \vert \ge \vert A \vert$, then necessarily $\vert I \vert \geq \operatorname{cof} \vert A \vert$