ThmDex – An index of mathematical definitions, results, and conjectures.

Result R7 on D13: Empty set

Empty set is subset of every set

Formulation 0

Let $X$ be a D11: Set.
Let $\emptyset$ be the D13: Empty set.

Then \begin{equation} \emptyset \subseteq X \end{equation}

Formulation 1

Let $X$ be a D11: Set.
Let $\emptyset$ be the D13: Empty set.

Then \begin{equation} \forall \, x \in \emptyset : x \in X \end{equation}

Formulation 2

Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$

Let $\emptyset$ be the D13: Empty set.

Then \begin{equation} \emptyset \in \mathcal{P}(X) \end{equation}

Proofs

Proof 0

Let $X$ be a D11: Set.
Let $\emptyset$ be the D13: Empty set.

If $X = \emptyset$, then the claim follows from R125: Subset relation is reflexive.

Consider then the case $X \neq \emptyset$. For $\emptyset$ not to be a subset of $X$ would require the existence of an element of $\emptyset$ that does not belong to $X$. If such an element were to exist, that would imply that $\emptyset$ is not empty, which is a contradiction. Hence, such element does not exist and therefore $\emptyset$ is a subset of $X$.