Indifference curves and equivalence classes
Yes, indifference is an equivalent relation.

That was covered in other answers and comments. I'm here to point out that preference relations are not partial orders. A partial order has the property that 

$$a\leq b, b\leq a \implies a=b$$

This not true precisely because of indifference. Rather, preference is typically assumed to be complete and transitive which means it is a total order on the quotient $X/\sim$