https://medlineplus.gov/ency/article/002465.htm#:~:text=A%20vegetarian%20diet%20does%20not,Vegetables

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Veganism is a type of Vegetarian

https://www.australianeggs.org.au/nutrition/eggs-and-vegetarians

You are not wrong that eggs can be considered Vegetarian, but you are also wrong saying that they are. Eggs and Cheese are in a gray area and it is diet, religious, and personal viewpoint on if they are actually vegetarian