Theorem 2.10: Let A and B be two sets. Then

Proof: Assume

To show that

let

by Theorem 1.2,

so

by Definition 1.1

For the converse, assume

To show that

we must show inclusion both ways by Definition 1.2. We already know that

by Theorem 1.2. For the reverse inclusion, let

then by Definition 1.3, either

in either event,

by Defnition 1.1, since we are assuming that