Subset of Empty Set

Theorem

Let $A$ be a class.

Then:

$A$ is a subset of the empty set $\O$

if and only if:

$A$ is equal to the empty set:

$A \subseteq \O \iff A = \O$

Proof

\(\ds A = \O\)

\(\leadsto\)

\(\ds A \subseteq \O\)

Definition 2 of Set Equality

Conversely:

\(\ds A \subseteq \O\)

\(\leadsto\)

\(\ds A \subseteq \O \land \O \subseteq A\)

Empty Set is Subset of All Sets

\(\ds \)

\(\leadsto\)

\(\ds A = \O\)

Definition 2 of Set Equality

$\blacksquare$