Concerning the cardinalities of sets...?

Question

How do I prove that:
if A, B, and C are non-empty sets,
A [is contained by] B [is contained by] C,
and the cardinality of A = the cardinality of C,

the cardinality of A = the cardinality of B

I can't figure out how to start this one.

Sorry, I should have specified these are indeed infinite sets.

Answer 1

If A is a subset of C and A has the same cardinality as C, then the sets A and C are the same.

I guess as a proof, you could assume that there is an element in C that is not in A...in which case the cardinality of C would be greater than the cardinality of A, which is a contradiction.

Then B gets trapped in the middle. A = B = C.

Answer 2

if Cardinal value of C is the same as the cardinal value of A it means the number of elements in C is the same as the no of elements in A,since C is a subset of A (given)
also the cardinality of B must be the same as the cardinality of C as C is also a subset of B
so by transitive property cardinality of B=cardinality of C

Answer 3

Well, if the sets are finite, it is easy. You know that |A|<=|B|<=|C|, so since |A|=|C|, you must have |A|=|B|=|C|.

If you're dealing with infinite sets, its a little harder. I'll get back to you in a minute.