Assume that P(A) = P(B). Show that A = B

Let P(A) = P(B)
To show: A = B
Let x ∈ A
P(A) = P(B) ⇒ x ∈ C, for some C ∈ P(B)
Now, C ⊆ B ⇒ x ∈ B
But x is an arbitrary element in A ⇒ A ⊆ B (1)
Now, let y ∈ B
P(B) = P(A) ⇒ y ∈ D for some D ∈ P(A)
D ⊆ A ⇒ y ∈ A
But y is an arbitrary element in B. Hence, B ⊆ A (2)
From (1) and (2), we get A = B