$X$ and $Y$ are equinumerous, written
\[X\equ Y,\]if there exists a bijection from $X$ onto $Y$.
$X$ is not a greater set than $Y$, written
\[X\lequ Y,\]if there exists an injection from $X$ to $Y$.
$X$ is a smaller set than $Y$, written
\[X\lnequ Y,\]if
\[X\lequ Y \enspace\land\enspace X\not\equ Y.\]
EQU#PROP-REF. Reflexivity of $\equ$.
For any $X$,
\[X \equ X.\]
EQU#PROP-SYM. Symmetry of $\equ$.
For any $X$ and $Y$,
\[X \equ Y \enspace\rimp\enspace Y \equ X.\]
EQU#PROP-TRA. Transitivity of $\equ$.
For any $X$, $Y$ and $Z$,
\[X\equ Y \,\land\, Y\equ Z \enspace\rimp\enspace X\equ Z.\]
$\equ$ is an equivalence relation.
Proof.By reflexivity, symmetry and transitivity of $\equ$.
EQU#PROP-L-REF. Reflexivity of $\lequ$.
For any $X$,
\[X \lequ X.\]
EQU#PROP-L-TRA. Transitivity of $\lequ$.
For any $X$, $Y$ and $Z$,
\[X \lequ Y \,\land\, Y\lequ Z \enspace\rimp\enspace X\lequ Z.\]
EQU#PROP-SB. Schröder-Bernstein Theorem.
For any $X$ and $Y$,
\[X \lequ Y \,\land\, Y\lequ X \enspace\rimp\enspace X\equ Y.\]
If $X\subseteq Y$, then $X\lequ Y$.
Proof.$\id_X:X\to Y$ is an injection.