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.
They answer to the first question is trivially yes, since every singleton is linearly ordered.
Revised to match edited question:
The answer to the second question is also trivially yes: let $\mathscr{D}=\{\{a\}:a\in A\}$.