Actually....Irvine wasn't a SeeD, he was just a skilled sharpshooter.

His non-SeeDness actually makes him feel alienated at various points in the game, which he at times expressed to the others, namely in the Galbadian mansion. It is easy to mistaken him for a SeeD but careful attention to the game's dialog reveals two things: 1. that he is never actually referred to as a SeeD and 2. that he himself at some point(s) expresses how "left out" he feels due to the others sharing a common purpose as SeeDs while he is not one.

Quistis, Squall, Selphie, and Zell are the only playable SeeDs in the game.