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

$\blacksquare$

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