Proof A⊂B if and only if A∩B = A

Show that A⊂B if and only if A∩B = A!
Proof :
Show that (A⊂B) ↔ (A∩B = A)
we have A⊂B,
it means x∈A, x∈B.

i. Show that A∩B ⊂ A
Take any x∈ A∩B
Obvious x∈A ∧ x∈B
↔ x∈A (simplifikasi)
We get for all x∈ A∩B, then x∈A
It means A∩B ⊂ A.......................1)

ii. Show that A ⊂ A∩B
Take any x∈ A
Obvious x∈A ∧ x∈A
↔ x∈A ∧ x∈B (because A⊂B,∀ x∈A, x∈B )
↔ x∈(A∩B)
We get for all x∈A, then x∈ A∩B
It means A ⊂ A∩B.......................2)

From (1) and (2) we conclude that A∩B = A is true,
So, if A⊂B then A∩B = A