I don't understand why $x \leq x \lor y$ and $y \leq x \lor y$ in the following basic lemma (taken from p.10, Models and Ultraproducts, Bell and Slomson (1969)):
Lemma 1.12: If the Lattice $L$ contains a maximal element then this maximal element is unique.
PROOF. Suppose that $x$ and $y$ are both maximal elements of $L$. Then $x \leq x \lor y$ and $y \leq x \lor y$ so that, since $x$ and $y$ are maximal, $x = x \lor y$ and $y = x \lor y$. Thus $x = y$ and the lemma follows.
From what elementary principle of lattices do $x \leq x \lor y$ and $y \leq x \lor y$ follow?
It's not from an elementary principle of lattices, so much as from the definition of $\vee$.
In a lattice (or any poset), $a\vee b$ ($a$ join $b$) is defined as the least upper bound of the elements $a$ and $b$. Of course, if $a\vee b$ is the least upper bound for the elements $a$ and $b$, it follows that it is an upper bound for them -- and therefore that $a\leq a\vee b$ and $b\leq a\vee b$.
The only thing that's special about a lattice here is that every two elements are guaranteed to have unique join and meet defined; you can actually take the definition of 'lattice' to be "a partially ordered set in which every pair of elements has a meet and a join".