What is the difference between $x:A$ and $x \Xi A$?
$x:A$ is a statement about objects in the formal system, like, for example, $\vdash 2+4:\texttt{int}$, whereas 
$x\Xi A$ is an expression in the formal system, like $\texttt{if}~ 2 + 4 ~\texttt{==}~5 ~\texttt{then}~ e_1~ \texttt{else} ~e_2$.