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:
 * $\forall A, B \subseteq S: A \cap B = \O \implies \map \rho {A \cup B} \le \map \rho A + \card B$

Proof

 * $\exists A, C \subseteq S : A \cap C = \O$ and $\map \rho {A \cup C} > \map \rho A + \card C$
 * $\exists A, C \subseteq S : A \cap C = \O$ and $\map \rho {A \cup C} > \map \rho A + \card C$

Let:
 * $A \subseteq S : \exists C \subseteq S : A \cap C = \O$ and $\map \rho {A \cup C} > \map \rho A + \card C$

Let:
 * $B \subseteq S : \card B = \min \set{\card C : C \subseteq S \land \map \rho {A \cup C} > \map \rho A + \card C}$

We have:

Hence:
 * $B \neq \O$

Let $y \in B$.

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

We have:

This contradicts the choice of $B$:
 * $\card B = \min \set{\card C : C \subseteq S \land \map \rho {A \cup C} > \map \rho A + \card C}$

It follows that:
 * $\forall A, B \subseteq S : A \cap B = \O \implies \map \rho {A \cup B} \le \map \rho A + \card B$