Is Veganism a subcategory of Vegetarianism?

Blobbenstein

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If a vegan is a subcategory of vegetarian, and so a vegan is a type of vegetarian, then a vegan should be able to eat food where the label says, 'suitable for vegetarians'.

Does the category depend on what foods are included, or what foods are excluded?
 
'Suitable for vegetarians' may include dairy or eggs, so not suitable for vegans.

It probably makes more sense to think of vegetarian as a subcategory of vegan (although I doubt this will catch on). Adding back foods (lacto, ovo, lacto-ovo) moves a step closer to omnivore.