Let $A$ be a countable set equipped with a partial order $\prec$. We all know that there are subsets $C\subset A$ with the property that the order $\prec$ in $C$ is total. In other words, any two elements $a$ and $b$ in $C$ are comparable.
Let $\mathscr{C}$ be the collection of all subsets $C$ of $A$ that are totally ordered.
Question 1: Is it true $A=\displaystyle\bigcup_{C\in\mathscr{C}}C$?
Denote by $\mathscr{D}$ a any subcolection $\mathscr{D}\subset \mathscr{C}$ such that for all $C_1,C_2\in\mathscr{D}$ we have empty intersection $C_1\cap C_2=\emptyset$.
Question 2: There is a subcollection $\mathscr{D}$ as described above such that $A=\displaystyle\bigcup_{C\in\mathscr{D}}C$?
I know these questions seem intuitive but I'd be happy with some demonstration of elementary set theory either by reduction to absurdity or not
Thank's.