The successor of an ordinal $\alpha$ is the ordinal
\[\alpha+1 = \alpha\cup\{\alpha\}.\]$\alpha$ is a successor ordinal, if $\alpha=\beta+1$ for some ordinal $\beta$.
For any ordinal $\alpha$,
\[\alpha+1 = \inf\{\beta:\beta>\alpha\}.\]
For any infinite ordinal $\alpha$,
\[\lvert\alpha+1\rvert = \lvert\alpha\rvert.\]
Proof.By PROP-EQU-SUB and Schröder-Bernstein Theorem, it is sufficient to show that $\lvert\alpha+1\rvert\leq\lvert\alpha\rvert$:
\[f : \alpha+1\to\alpha, \, \xi\mapsto\begin{cases} 0, & \text{if $\xi=\alpha$}; \\ \xi+1, & \text{if $\xi<\omega$}; \\ \xi, & \text{otherwise} \end{cases}\]is an injection.