Uniform Matroid is Matroid

Theorem
Let $S$ be a finite set of cardinality $n$.

Let $0 \le k \le n$.

Let $U_{k, n} = \struct{S, \mathscr I}$ be the uniform matroid of rank $k$.

Then $U_{k, n}$ is a matroid.

Proof
It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(I1)$, $(I2)$ and $(I3)$.

Matroid Axiom $(I1)$
Now:

By definition of the uniform matroid of rank $k$:
 * $\O \in \mathscr I$

It follows that $\mathscr I$ satisfies matroid axiom $(I1)$.

Matroid Axiom $(I2)$
Let $X \in \mathscr I$.

Let $Y \subseteq X$.

Then:

By definition of the uniform matroid of rank $k$:
 * $Y \in \mathscr I$

It follows that $\mathscr I$ satisfies matroid axiom $(I2)$.

Matroid Axiom $(I3)$
Let $U, V \in \mathscr I$.

Let $\size V < \size U$.

From Intersection is Subset:
 * $U \cap V \subseteq V$
 * $U \cap V \subseteq U$

Then:

Then:
 * $U \cap V \ne U$

It follows that:
 * $U \cap V \subsetneqq U$

Then:

Then:

By definition of the uniform matroid of rank $k$:
 * $V \cup \set x \in \mathscr I$

It follows that $\mathscr I$ satisfies matroid axiom $(I3)$.