Matroid closure exchange property ($y\notin {\mathrm{cl}}_M(A),y\in {\mathrm{cl}}_M(A+x)\Rightarrow x\in {\mathrm{cl}}_M(A+y)$)

The closure of a matroid obeys the displayed exchange rule. This is the finite pregeometry exchange principle: if adding $x$ newly makes $y$ dependent, then adding $y$ also makes $x$ dependent.

Proof. Write $r=r(A)$. The first hypothesis and the one-element bound on the rank of a matroid give $r(A+y)=r+1$. The second gives $r(A+x+y)=r(A+x)\le r+1$. Inclusion gives $r(A+x+y)\ge r(A+y)$, so equality holds. This is exactly $x\in {\mathrm{cl}}_M(A+y)$.