Actually, now that I think about, isn't

$\cap_X(x,x') =\begin{cases}\mathrm{true} & \mathrm{if} \ x \le x' \\\ \mathrm{false} & \mathrm{otherwise} \end{cases}$

a bit redundant? \$$[x \leq x']\$$ is already *the* relation that gives \$$\mathrm{true}\$$ when \$$x\$$ is less then or equal to \$$x'\$$, and \$$\mathrm{false}\$$ otherwise. Running a conditional on \$$[x \leq x']\$$ to give either \$$\mathrm{true}\$$ or \$$\mathrm{false}\$$ is therefor redundant.

Or to be blunter,

\$\cap_X(x,x') := [x \leq x']. \$