I have seen various proofs (see, for example, here: https://math.stackexchange.com/a/436074/264885) where we're saying if $a$ is not a maximal element, then there necessarily is some $b$ such that $a < b$. My question is, why is this true? How can we assume $a$ and $b$ are comparable because by my understanding of the poset axioms any two elements need not be comparable.
Also, I'm using the following definition of a maximal element: "Let $(P,\leq)$ be a partially ordered set and $S\subset P$. Then $m\in S$ is a maximal element of $S$ if for all $s\in S, m \leq s$ implies $m = s$."