Equivalence of Definitions of Matroid Rank Axioms/Lemma 1

Theorem
Let $S$ be a finite set.

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.

Let $\rho$ satisfy the rank axioms:

Then:
 * $\forall A, B \subseteq S: A \subseteq B \implies \map \rho A \le \map \rho B$

Proof

 * $\exists A, B \subseteq S : A \subseteq B$ and $\map \rho A > \map \rho B$
 * $\exists A, B \subseteq S : A \subseteq B$ and $\map \rho A > \map \rho B$

Let $B \subseteq S$:
 * $\exists A \subseteq B : \map rho A > \map \rho B$

Let $A_0 \subseteq B$:
 * $\card {A_0} = \max \set{\card A : A \subseteq B \land \map \rho A > \map \rho B}$

As $\map \rho {A_0} > \map \rho B$:
 * $A_0 \ne B$

From Set Difference with Proper Subset:
 * $\exists y \in B \setminus A_0$

We have:

This is a contradiction.

Hence:
 * $\forall A, B \subseteq S: A \subseteq B \implies \map \rho A \le \map \rho B$