Empty Set is Submagma of Magma

Theorem
Let $\struct {S, \circ}$ be a magma.

Then:
 * $\struct {\O, \circ}$ is a submagma of $\struct {S, \circ}$

where $\O$ is the empty set.

Proof
By definition, a magma is an algebraic structure $\struct {S, \circ}$ where $\circ$ is closed.

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

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

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

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

Hence the result.