Which Came First, The Milk Or The Egg?

(At school, I am taking an advanced cooking and catering class, and we need to write grocery orders for our teacher on what ingredients we need. We are sorting the ingredients based on what section you might find in the grocery store, and one of my group members is VERY confused. Our conversation goes a little something like this.)

Classmate: “So, mayonnaise is a dairy product, right?”

Me: “No, it’s a condiment. Besides, it’s made from eggs.”

Classmate: “But eggs are dairy!”

Me: “Eggs are not dairy.”

Classmate: “But they’re found in the dairy section!”

Me: *opens lunchbox, takes out coconut milk-based vegan yogurt* “So’s this, but that doesn’t make it dairy. I’ll even Google it for you.” *Googles “Are eggs dairy” and pulls it up to show classmate, obviously saying that eggs are NOT dairy* “You realize dairy means milk products, right?”

Classmate: “Yeah.”

Me: “Eggs are not made from milk.”

Classmate: “But they’re made from milk-producing animals, so they’re classified as dairy.” *he is talking to me in a tone as though I am really stupid*

Me: “You realize you basically just said that cows lay eggs, right?”

Classmate: *says nothing, writes “mayonnaise” in the dairy section anyway*

(We are basically yelling this conversation, but nobody else in the classroom is paying attention. We continue shouting at each other about eggs for a few more minutes, the other members of our group looking very confused. I walk over to my teacher:)

Me: “I’ve been trying to teach [Classmate] that eggs aren’t dairy, but he still thinks that mayonnaise is a dairy product. Maybe you can help him?”

Teacher: “It doesn’t even matter! Mayonnaise is a condiment!”

Me: “That’s what I said!”

Another Classmate: *yelling* “But eggs ARE dairy!”

(You know, I’d think high-schoolers in an advanced-level cooking class would know this stuff.)