is this a legal expression in math? (set theory)

Question

$\mathcal{N^{(i)}}$={$\mathcal(m,s,k)$$\in$$\mathcal{U^{(i)}}$:$\exists$$\mathbf{x}$$\in$$\mathcal{E_{(m,s,k)}}$:$\mathbf{x}$$\notin$$\mathcal{V}$}

$\mathcal{E_{(m,s,k)}}$ is a set related to (m,s,k)

and i want to find tuples(m,s,k) such that $\exists$$\mathbf{x}$$\in$$\mathcal{E_{(m,s,k)}}$:$\mathbf{x}$$\notin$$\mathcal{V}$

is it possible to use : notation twice?

Answer 1

I believe it's frowned upon. It would be better to reserve "$:$" for set builder notation and use parentheses with quantifiers, as for example $$\{x: \exists y(x\oplus y\in Q)\}.$$

Now in your case that expression can only be parsed in one way, so it's not too bad, but I do think it tends to impair readability.

Answer 2

I agree with Noah's answer but would like to suggest an alternative:

Like you, I like to use colons to separate quantifiers (and their variables) from the rest of the formula. For that reason, I use $|$ in set builder notation. Like this: $$ \{x \ | \ \exists y \colon x\oplus y\in Q\}. $$