Empty Set as Subset
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Theorem
Let $S$ be a set.
Let $A$ be a subset of $S$.
Then:
- $A = \O \iff \forall x \in S: x \notin A$
Proof
Sufficient condition follows by definition of empty set.
For necessary condition assume that:
- $\forall x \in S: x \notin A$
Let $x$ be arbitrary.
Aiming for a contradiction, suppose that:
- $x \in A$
By definition of subset:
- $x \in S$
By assumption:
- $x \notin A$
Thus this contradicts:
- $x \in A$
$\blacksquare$