Empty Set is Submagma of Magma

Theorem
Let $\left({S, \circ}\right)$ be a magma.

Then:
 * $\left({\varnothing, \circ}\right)$ is a submagma of $\left({S, \circ}\right)$

where $\varnothing$ is the empty set.

Proof
By definition, a magma is an algebraic structure $\left({S, \circ}\right)$ where $\circ$ is closed.

That is:
 * $\forall x, y \in S: x \circ y \in S$

By definition, $\left({T, \circ}\right)$ is a submagma of $S$ if:
 * $\forall x, y \in T: x \circ y \in T$

But:
 * $\not \exists x, y \in \varnothing: x \circ y \notin \varnothing$

it follows vacuously that:
 * $\forall x, y \in \varnothing: x \circ y \in \varnothing$

Hence the result.