Equivalence of Definitions of Cardinality of Finite Class

Proof
Let $A_1$ be the class which has a bijection $\phi_1$ from $A_1$ to $n$.

Let $A_2$ be the class which has a bijection $\phi_2$ from $A_2$ to $n^+ \setminus \set 0$.

Consider the mapping $\phi: A_1 \to A_2$ defined as:
 * $\forall k \in n: \map {\phi_1} k = k^+$

$\phi$ is trivially a bijection.

The result follows.