Equivalence of Definitions of Matroid Rank Axioms/Lemma 2

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:
 * $\forall A \subseteq S: \map \rho A \le \card A$

Proof

 * $\exists A \subseteq S : \map \rho A > \card A$
 * $\exists A \subseteq S : \map \rho A > \card A$

Let:
 * $A_0 \subseteq S : \card{A_0} = \min \set{\card A : \map \rho A > \card A}$

We have:

Hence:
 * $A_0 \neq \O$

Let $y \in A_0$.

From Cardinality of Set Difference:
 * $\card {A_0 \setminus \set y} = \card{A_0} - 1 < \card {A_0}$

We have:

This contradicts the choice of $A_0$:
 * $\card{A_0} = \min \set{\card A : \map \rho A > \card A}$

It follows that:
 * $\forall A \subseteq S: \map \rho A \le \card A$