Subset of Empty Set

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Theorem

Let $A$ be a class.

Then:

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

if and only if;

$A$ is equal to the empty set:
$\displaystyle A \subseteq \varnothing \iff A = \varnothing$


Proof

\(\displaystyle A = \varnothing\) \(\implies\) \(\displaystyle A \subseteq \varnothing\) Definition of Set Equality


Conversely:

\(\displaystyle A \subseteq \varnothing\) \(\implies\) \(\displaystyle A \subseteq \varnothing \land \varnothing \subseteq A\) Empty Set is Subset of All Sets
\(\displaystyle \) \(\implies\) \(\displaystyle A = \varnothing\) Definition of Set Equality

$\blacksquare$