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Mario Tsai in Valcucine Milano. By Mario Tsai Studio.

Example:

"I am agender, but don't want to be called non-binary" - thats fine. You don't have to call yourself non-binary.

However, when you say "I am agender and I don't like to call myself non-binary. Therefore, being agender doesn't fall under the non-binary umbrella, and if you say it does, you are being bigoted" is completely illogical.

Agender is a lack of gender. The non-binary umbrella includes any gender (or lack thereof) that is not exclusively monogendered woman or monogendered man. Therefore, agender falls under the non-binary umbrella.

You do not have to identify as non-binary personally, but the umbrella term still includes agenderness because it fits under the definition of the umbrella.

Another example:

"I have PCOS, but don't want to be called intersex" - again, fine. But "I have PCOS, and I don't want to call myself intersex, therefore PCOS doesn't fall under the intersex umbrella" is illogical.