Reflection principle: can any property of a set (model) be reflected?

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Can any property of a set be reflected? If no, why? Is there some sort of requirement that must be fulfilled? I've read on the Wikipedia page https://en.m.wikipedia.org/wiki/Reflection_principle ,that in ZF the property of the 'class of all sets' being reflected causes some paradoxical implications. I am relatively new to set theory and as of recently interested in the reflection principle. I would like questions about it answered, and any misconceptions cleared.