[I'll use 'E' for 'there exists', 'e' for 'is an element of' and 'A' for 'for all']

Ex(xev & Ay(yev -> y=x))

By the way, you probably meant not

{aeX | P[v]}

but rather

{veX | P[v]}

where the variable 'X' does not occur free in P[v]

or

{aeX \ P[a]}

where the variable 'X' does not occur free in P[v]