Prove = is an equivalence relation for sets

For any sets A,B, C Show that '=' has the following properties
(i) Reflexive. A=A.
(ii) Symmetric. If A=B, then B=A.
(iii) Transitive. If A=B and B=C then A=C.

(i)Proof: '=' is defined to mean that two sets have the exact same elements. Clearly A has the same elements as itself.
(ii)Proof: For all x in A and for all y in B A=B implies that
all y are in A and all x are in B. Thus B=A as well.
(iii)Proof: For all x in A, y in B, and z in C
A=B implies that all y are in A
and
all x are in B.
B=C implies
all z are also in B and all y are in C.
But we know that all y are also in A since A=B.
This A=C.

I know these are basic proofs. they hard part is getting around the fact
that they are so intuitively true.
Are my proofs correct?
.