User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Lemma

Theorem
Let $S$ be a finite set. Let $B_1, B_2 \subseteq S$.

Let $x \in B_1 \setminus B_2$.

Let $y \in B_2 \setminus B_1$.

Then:
 * $\paren{B_1 \setminus \set x} \cup \set y = \paren{B_1 \cup \set y} \setminus \set x$

Proof
From Singleton of Element is Subset:
 * $\set x \subseteq B_1 \setminus B_2$

and
 * $\set y \subseteq B_2 \setminus B_1$

From Set Difference is Disjoint with Reverse:
 * $\paren{B_1 \setminus B_2} \cap \paren{B_2 \setminus B_1} = \O$

From Subsets of Disjoint Sets are Disjoint:
 * $\set x \cap \set y = \O$

We have: