$X$ is a subset of $Y$, written $X\subseteq Y$, if
\[\forall x :\enspace x\in X \enspace\rimp\enspace x\in Y.\]$X$ is a proper subset of $Y$, written $X\subset Y$, if $X\subseteq Y$ and $X\neq Y$.
$\varnothing$ is the only set that is a subset of any non-empty set.
Proof.$\varnothing$ is a subset of any set since no $x\in\varnothing$. If $X$ is a subset of any non-empty set, then $X\subseteq\{X\}$. By PROP-ZF-SLF, $X\neq\{X\}$, so $X=\varnothing$.
For any $X$,
\[X\subseteq X.\]
For any $X$, $Y$ and $Z$,
\[X\subseteq Y\land Y\subseteq Z \enspace\rimp\enspace X\subseteq Z.\]
For any $X$ and $Y$,
\[X\subseteq Y\land Y\subseteq X \enspace\rimp\enspace X=Y.\]
As a result:
$\subseteq$ is a partial order.