I think your "exists x" inside parentheses in the last two steps are typos (or if not.. then they are mistakes).
Exercise 1(e) in Section 2.2 of Daniel Velleman's How to prove it:
Negate the statement and then reexpress the result as an equivalent positive statement:
My attempt at the solution is by expressing it in the logical form as
Step 2: negate the statement
Question: Is there a set notation that is equivalent to the last line?