The Invertible Matrices and the Positive Semidefinite Matrices Form Sets

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n\in\mathbb{N}$, with the square real matrices $\mathcal{M}_{n}(\mathbb{R})$, the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, invertible matrices as in Inverse Matrix and Invertible Real Square Matrix §invertible, and positive semidefinite matrices as in Positive Semidefinite and Positive Definite Real Matrices §psd.

¶ There is exactly one set $\mathrm{GL}_{n}(\mathbb{R})$ such that, for every set $A$, $A\in\mathrm{GL}_{n}(\mathbb{R})$ if and only if $A\in\mathcal{M}_{n}(\mathbb{R})$ and $A$ is invertible. It is a subset of $\mathcal{M}_{n}(\mathbb{R})$, written $\{A\in\mathcal{M}_{n}(\mathbb{R}):A\text{ is invertible}\}$.

¶ There is exactly one set $\mathcal{S}_{+}(n)$ such that, for every set $A$, $A\in\mathcal{S}_{+}(n)$ if and only if $A\in\mathcal{S}(n)$ and $A$ is positive semidefinite. It is a subset of $\mathcal{S}(n)$, written $\{A\in\mathcal{S}(n):A\text{ is positive semidefinite}\}$.