Rank Function is Increasing

Theorem
Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\rho : \powerset S \to \Z$ be the rank function of $M$.

Let $A, B \subseteq S$ be subsets of $S$ such that $A \subseteq B$.

Then:
 * $\map \rho A \le \map \rho B$

Proof
Now:

Similarly:

From Power Set of Subset:
 * $\powerset A \subseteq \powerset B$

From Set Intersection Preserves Subsets:
 * $\powerset A \cap \mathscr I \subseteq \powerset B \cap \mathscr I$

It follows that:
 * $\set {\size X : X \in \powerset A \cap \mathscr I} \subseteq \set {\size X : X \in \powerset B \cap \mathscr I}$

From Max of Subfamily of Operands Less or Equal to Max:
 * $\max \set {\size X : X \in \powerset B \cap \mathscr I} \le \max \set {\size X : X \in \powerset B \cap \mathscr I}$