What is wrong with this proof of finite sets?

Question

Let A, B, C, D be finite sets with A⊆C and B⊆D. What is wrong with the following incorrect proof that A×B⊆C×D.

Proof:

Since A⊆C we have that |A|≤|C|. Since B⊆D we have that|B|≤|D|.

We know that |A×B|=|A| × |B| and |C×D|=|C|×|D| .So |A×B| = |A|×|B| ≤ |C|×|D| = |C×D|.

Since the cardinality of A×B is less that the cardinality of C×D, we must have that A×B is a subset of C×D

Answer 1

The cardinality and the relation of subset between sets are different concepts. If A is a subset of B then it is true that the card(A)<=card(B). But the converse is not true. For example if A={0,1} and B={2,3,4} then card(A)=2<card(B)=3 but A is NOT a subset of the set B.