Such that in set-builder notation

Question

I want to define a set $A=\{x:\exists y\ni P(x,y)\}$. That is, $A$ is the set of all $x$ such that there exists a $y$ such that $P(x,y)$ holds. Is using $\ni$ fine, or should I write $A=\{x:\exists y:P(x,y)\}$ or $A=\{x:\exists y, P(x,y)\}$? Thanks.