[Tex/LaTex] Alignat with “mixed” left-right alignment

Question

Here is some LaTeX fragment, which requires importing amsmath and amssymb:

\begin{alignat}{2}
& \forall \left\{i_1, i_2\right\} \in \binom{\left\{1, \ldots, t\right\}}{2}: \mathcal{U}\left(s_{i_1}, b_{i_1}\right) \cap \mathcal{U}\left(s_{i_2}, b_{i_2}\right) = \emptyset \nonumber \\
\Leftrightarrow & \forall \left(\left(i_1, j_1\right), \left(i_2, j_2\right)\right) \in \mathcal{I}: 
\left(a_{i_2} z_{i_2} + \sigma_{s_{i_2}, j_2}\right) - 
\left(a_{i_1} z_{i_1} + \sigma_{s_{i_1}, j_1} + m_{j_1}\right) && \geq 0
\nonumber \\
& \vee
\left(a_{i_1} z_{i_1} + \sigma_{s_{i_1}, j_1}\right) - 
\left(a_{i_2} z_{i_2} + \sigma_{s_{i_2}, j_2} + m_{j_2}\right) && \geq 0. \label{eq:disj}
\end{alignat}

Here is how the result looks like (the number of the equation is (7) instead of (1) since it is a screenshot from a document that contains some more text and formulas).

And here is how I wished that it looked like (first half of the last inequality right-aligned instead of left-aligned):

So the question is obvious: How does not need to change the source code to get the desired result?

Answer 1

A solution with mathrlap from mathtools (needless to load amsmath). I removed almost all unnecessary \left \right pairs, and replaced the remaining one with a pair \bigl \bigr. I also took the liberty to replace \emptyset with the more correct \varnothing:

\documentclass[11pt]{article}
\usepackage{geometry} 
\usepackage{amssymb, mathtools}

\begin{document}

\begin{alignat}{2}
& \forall\mathrlap{\{i_1, i_2\} \in \binom{\{1, \ldots, t\}}{2} : \mathcal{U}(s_{i_1}, b_{i_1}) \cap \mathcal{U}(s_{i_2}, b_{i_2}) = \varnothing} \nonumber & \\
\Leftrightarrow {}& \forall \bigl((i_1, j_1), t(i_2, j_2)\bigr) \in \mathcal{I}&:
(a_{i_2} z_{i_2} + \sigma_{s_{i_2}, j_2}) -
(a_{i_1} z_{i_1} + \sigma_{s_{i_1}, j_1} + m_{j_1})& \geq 0
\nonumber \\
& & \vee
(a_{i_1} z_{i_1} + \sigma_{s_{i_1}, j_1}) -
(a_{i_2} z_{i_2} + \sigma_{s_{i_2}, j_2} + m_{j_2})& \geq 0. \label{eq:disj}
\end{alignat}

\end{document}