Let A, B, and C be the sets such that A∪B = A∪C and A∩B = A∩C. Show that B = C

Let, A, B and C be the sets such that A∪B = A∪C and A∩B = A∩C.
To show: B = C
Let x ∈ B ⇒ x ∈ A∪B (By def of union of sets) ⇒ x ∈ A∪C (A∪B = A∪C) ⇒ x ∈ A or x ∈ C
Case I: x ∈ A
Also, x ∈ B ∴ x ∈ A∩B ⇒ x ∈ A∩C (∵ A∩B = A∩C) ∴ x ∈ A and x ∈ C ∴ x ∈ C
But x is an arbitrary element in B. ∴ B ⊆ C (1)
Now, we will show that C ⊆ B.
Let y ∈ C ⇒ y ∈ A∪C (by def of union of sets) ⇒ y ∈ A∪B (A∪B = A∪C) ⇒ y ∈ A or y ∈ B
Case I : When y ∈ A
Also, y ∈ C ⇒ y ∈ A∩C ⇒ y ∈ A∩B ⇒ y ∈ A and y ∈ B ⇒ y ∈ B
But y is an arbitrary element of C. Hence, C ⊆ B (2)
From (1) and (2), we get ∴ B = C.