Indifference curves and equivalence classes Last week in class, my professor described indifference curves in the context of modelling consumer choice with multiple goods. He defined a partial-ordering operator $\prec$ and an equivalence operator $\sim$ to manipulate tuples that satisfied the budget constraint problem. So far, so good. 

But then we went on to explore the behavior of indifference curves and I noticed that the $\sim$ was assumed to be reflexive, symmetric, and transitive. We learnt that indifference curves never intersect and that there is a completeness property that says that for all the tuples in our domain space belong to an indifference curve.

At this point, indifference curves looks like textbook equivalence classes to me, but I would like confirmation and perhaps some context from an economics perspective.

First of all, why not call them "equivalence classes" or openly saying that "indifference curves are equivalence classes". Second, does that mean that if I prove something about equivalence classes, then it automatically transfers to indifference curves in the context of this economic model?
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 $X/\sim$