I have an explanation here as to why the GLB of two elements in a poset is unique whenever it exists... but I can't quite understand the concept of GLB and LUB.
Suppose $x$ and $y$ are each glbs of two elements $a$ and $b$. Then $x\preceq a$, $x\preceq b$ implies $x\preceq y$ because $y$ is a greatest lower bound, and $y\preceq a$, $y\preceq b$ implies $y\preceq x$ is greatest. So by antisymmetry, $x=y$.
Looking at this, I have to ask why $x\preceq a\wedge x\preceq b\Rightarrow x\preceq y$.