Theorem: Suppose $\A$ and $\B$ are positive definite matrices satisfying $\A \gg \B$. Then $\A^{-1} \glt \B^{-1}$.
Proof: Consider the partitioned matrix
\[\M = \begin{bmatrix} \A & \I \\ \I & \B \end{bmatrix}.\]Noting that $\M$ is positive definite if $\A \gg \B$, we can conclude that $(\M^{-1})_{22}$ is also positive definite (as is its inverse). Using the Schur complement, we therefore have
\[\B^{-1} - \I\A^{-1}\I = \B^{-1}-\A^{-1} \gg \zero\]