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The anime works like this. Basically not having magic means death and only those with magic can live.

The scene I saw is a princess in that city secretly doesn't have magic and is later found out (she's not the main character).

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Might this be Cross Ange: Tenshi to Ryuu no Rondo (2014–2015)...?

From Wikipedia:

Through the development of groundbreaking information transfer and material generation technology called "Mana", various problems, such as pollution and war have disappeared, and days of peace have arrived in the world. However, the people who receive the benefit of the Mana live in a false peace. Those who cannot use the Mana, referred to by the derogatory term "Norma", who are seen as an opposition to the structures of society, live in oppression. Because this is the norm, the people who use Mana do not question it.

The first princess of the Misurugi Empire, Angelise Ikaruga Misurugi, was convinced of the belief of exterminating the Norma. But on the day she turned 16, and was baptised, her elder brother, Julio, exposed her as a Norma to the public. Angelise, bewildered by the revelation that she herself was a Norma all along, turns into a target of hate by the people in the blink of an eye and is exiled to Arzenal, an isolated military base where Norma are conscripted into service against their will to fight interdimensional creatures known as DRAGONs, and starts her new life as the Norma soldier named Ange.

As she struggles to adjust to her new identity, Ange's now stormy life takes one surprising turn after another as she embarks on a personal quest to correct the world she lives in, all while learning there may be a bigger conspiracy involving the DRAGONs, the world she lives in, and mana itself.

This article from Kotaku talks a bit more about the prejudice against non-magic users, and the consequences of that for the princess.

Cross Ange is the story of Angelise, the princess of a futuristic kingdom where everyone can use magic — well, almost everyone. In this world, people who can’t use magic, and are therefore immune to it, are treated as inhuman abominations to be rounded up and imprisoned far away from normal society. But more than that, they are actually conscripted and forced to pilot giant mecha and kill dragons in order to protect the utopian society that has imprisoned them. Unfortunately for Angelise, while she may be a princess, she is also a non-magic user (though this fact was cleverly hidden from her by her parents). And when this is exposed, she is treated as any other non-magical human and is thrown in prison to battle dragons until she is inevitably killed.

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    @Justsomerandomguy - If anyone correctly identifies the anime you're looking for, please mark that answer as accepted by clicking on the check mark beneath the voting buttons. Or if the only answers posted are incorrect, please indicate that with a comment so we know to keep looking. Thanks. Nov 11 '21 at 12:22
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    This differs from OP's recollection in that Angelise is the main character, but from what I can recall, Cross Ange also has quite a large ensemble cast, so it's reasonable to assume that OP may have misremembered. (Also, trigger warning for anyone who wants to watch this show: it contains multiple scenes of rape and sexual assault within the first two episodes (which was all I could stand to watch of it)).
    – F1Krazy
    Nov 11 '21 at 12:26

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