Two sets $X$ 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 $X \lequ Y$ and $X \not\equ Y$.
Proposition 1. Reflexivity of $\equ$.
For any set $X$,
\[X \equ X.\]
Proposition 2. Symmetry of $\equ$.
For any sets $X$ and $Y$,
\[X \equ Y \enspace\rimp\enspace Y \equ X.\]
Proposition 3. Transitivity of $\equ$.
For any $X$, $Y$ and $Z$,
\[X \equ Y \,\land\, Y \equ Z \enspace\rimp\enspace X \equ Z.\]
As a corollary of reflexivity, symmetry and transitivity of $\equ$:
For any set $X$, $\equ$ is an equivalence relation on $\powerset(X)$.
Proposition 5. Reflexivity of $\lequ$.
For any set $X$,
\[X \lequ X.\]
Proposition 6. Transitivity of $\lequ$.
For any sets $X$, $Y$ and $Z$,
\[X \lequ Y \,\land\, Y\lequ Z \enspace\rimp\enspace X\lequ Z.\]
Proposition 7. Schröder-Bernstein Theorem.
For any sets $X$ and $Y$,
\[X \lequ Y \,\land\, Y\lequ X \enspace\rimp\enspace X\equ Y.\]
For any sets $X$ and $Y$, if $X \subseteq Y$, then $X \lequ Y$.
Proof. $\id_X : X \to Y$ is an injection.