Theorem 2.11: Let A and B be two sets. Then

Proof: Assume

Since

by Theorem 1.3, it follows that

by Theorem 2.7. For the converse, assume

To show that

we must show inclusion both ways by Definition 1.2. We get

from Theorem 1.3. For the reverse inclusion, let

Then, since

we also have

by Definition 1.1. Then

by Definition 1.4. Thus

Since we havbe inclusion both ways we get that

by Definition 1.2.