x and y are sets.Show that for any x there exists a transitive y such that

Here's my idea: Take a set x and take it's cardinality. Express this number in terms of power sets of the empty set. This would give a transitive set every time.

The only problem is if the set we start off with is in terms of power sets of the empty set. In this case, map this to the cardinality of it's set +1 and express that number in terms of power sets of the empty set.

Is this a good way of proving this?