Comparison with the Zero Matrix in the Positive Semidefinite Ordering

Let $n$ be a natural number, let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices, and let $A$ and $B$ belong to $\mathcal{S}(n)$.

Let $0_n$ denote the real $n\times n$ matrix all of whose entries are $0$. Its entries satisfy $(0_n)_{ij}=(0_n)_{ji}$, so $0_n$ is symmetric and lies in $\mathcal{S}(n)$, and by The Positive Semidefinite Ordering Compared by Differences the difference $A-B$ is symmetric and lies in $\mathcal{S}(n)$ as well. The relation $\preceq$ below is the positive semidefinite ordering.

Then the following hold.

1. (Comparison from above) $A-B\preceq 0_n$ if and only if $A\preceq B$.

2. (Comparison from below) $0_n\preceq A-B$ if and only if $B\preceq A$.