In a mathematical context: Let's say x is contained in an interval I. How can I express this as a noun equivalent (but correct) to the "containedness"? I found things like "inherence", "immanence", but they don't seem to fit perfectly
Speaking in mathematical terms, inclusion sounds like a good candidate.
In your example, you could be writing a "proof of inclusion of x in I"
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I am a set theorist and have so far found no satisfactory way of expressing this: I don't believe there is any adjective to describe "being a subset of" that couldn't also mean "being an element of." "Inclusion of x in I" doesn't cut it, for example; that could mean that x is an element of I. Same with "containment" and the verb-phrase "is contained in." Unless it's totally clear from context, I'd use "is an element of" or "is a subset of" always.