Properties of indifference curve:
Property I: Higher indifference curve gives higher utility.
Explanation: Since all goods are non-satiated, larger consumption of any good leads to larger utility. Thus, a commodity bundle, which consists of larger quantity of at least one good and no less consumption of any other goods, gives larger utility compared to any other commodity bundle. Consequently, higher indifference curve represents higher consumption of at least one commodity and no less consumption of any other commodity.
Property II: Indifference curves can't intersect with each other.
Explanation: Suppose two indifference curves intersect each other. By definition, along the indifference curve, utility is constant. So, consumer is indifferent between points 'A' and 'C' that lie on the same indifference curve. Similarly, consumer is indifferent between points 'B' and 'C', as they also lie on the same indifference curve. So, AIC and BIC, where 'I' denotes indifference. Now, from transitivity we have AIB i.e., point 'A' and point 'B' give the same utility to the consumer. But for given x_{2}, x_{1} is larger in point 'A' compared to point 'B'. So, by the assumption of non-satiation, we have point 'A' that gives lager utility to consumer as compared to point 'B'. This contradicts the fact that point 'A' and 'B' gives the same level of utility to the consumer (as we have proved above). Therefore, when all goods are non-satiated and transitivity holds, indifference curves can't intersect.