For my work I need many of the very easy and basic properties of suprema and infima. While they are all pretty easy to prove, I would prefer to refer to a standard text book. However I did not find one and worse, do not know how to search for one.
I am interested in properties for partially ordered sets like
$A\subseteq B \Rightarrow \inf B \le \inf A$
$\inf \bigcup A=\inf \{\inf a|a\in A\}$
And for sets with a "$+$", and an inverse $-$ (i.e. $-x \le -y \leftrightarrow x\ge y$):
$\inf X=-\sup (-X)$
$\inf (X+m) = (\inf X) +m $
And the equivalent for functions into such sets ($f: Y \rightarrow X$):
$\inf_{y\in Y} f(y)=-\sup_{y\in Y} -f(y)$
and so on...
Is there like a "reference handbook"?