I'm quite familiar with the pigeonhole principle (strong form), namely that principle which says that if $q_1 + q_2 + ... + q_n – n + 1$ objects are distributed (or taken) to (or from) $n$ boxes, then for some $i = 1, 2, ..., n$, it is the case that the $i$th box contains $q_i$ objects (or that $q_i$ objects were taken from the $i$th box. However, for some reason, I've only seen this being stated with boxes; and I believe it could be equivalently restated using actual mathematical objects. This is how I attempted this restatement:
if $\{X_i\}_{i \in I}$ is a family, $I = \{1, 2, ..., n\}$ and $\vert \{X_i\}_{i \in I}\vert = q_1 + q_2 + ... + q_n – n + 1$,then $(\exists i \in I)(\exists X_i \in \{X_i\}_{i\in I})(X_i \ge q_i)$
Is this a correct restatement of the fact? I am nothing but new to combinatorics, hence I need approval from those more professional. Moreover, is there a traditional restatement of this principle with mathematical objects? I would be most interested in that. Thank you in advance.