If A is any set, prove that:

To prove: $A \subseteq \phi \Leftrightarrow A=\phi$

Proof:

Let:

$A \subseteq \phi$

If A is a subset of an empty set, then A is the empty set.

$\therefore A=\phi$

Now, let $A=\phi$

This means that A is an empty set.

We know that every set is a subset of itself.

$\therefore A \subseteq \phi$

Thus, we have:

$A \subseteq \phi \Leftrightarrow A=\phi$