Question

If A is any set, prove that: $A\subseteq \varphi \iff A=\varphi .$

Answer 1

To prove: $A\subseteq \mathrm{\varphi }\iff A=\mathrm{\varphi }$
Proof:
Let:
$A\subseteq \varphi$
If A is a subset of an empty set, then A is the empty set.
∴ $A=\varphi$

Now, let $A=\varphi$.
This means that A is an empty set.
We know that every set is a subset of itself.
∴ $A\subseteq \varphi$
Thus, we have:
$A\subseteq \varphi \iff A=\varphi$