I work here in $\mathrm{ZFC}$. Transfinite induction is the following observation:
Let $P(x)$ be a property. If
$(1) \ P(0)$ is true
$(2)$ for any ordinal $\alpha:$ $P(\alpha)$ is true $\Rightarrow P(\alpha + 1)$ is true
$(3)$ for any limit ordinal $\alpha$: $P(\beta)$ is true for all $\beta < \alpha \Rightarrow P(\alpha)$ is true
Then $P(\alpha)$ is true for all ordinals $\alpha$
Then one definition of ordinal addition is the following:
Let $\alpha$ be an ordinal.
$(1) \ \alpha + 0 = \alpha$
$(2) \ \forall$ ordinals $\beta: \alpha + (\beta + 1) = (\alpha + \beta) + 1$
$(3) \forall$ limit ordinals $\beta: \alpha + \beta = \mathrm{sup}\{ \alpha + \gamma \ | \ \gamma < \beta \}$
How can we actually make sure that the preceding definition of ordinal addition is actually... a definition? That it makes sense? Probably, rigorously it means that we need to shows that $\alpha_1 = \alpha_2, \beta_1 = \beta_2 \Rightarrow \alpha_1 + \beta_1 = \alpha_2 + \beta_2$. Which can probably be attempted by transfinite induction. However, we also need to make sense of $\{ \alpha + \gamma \ | \gamma < \beta \}$. It should be proved that it is even a set.
Here lies the problem. If one tried to first prove that that definition for addition defines the unique "sum", then one would stumble on the fact that we don't know if $\{ \alpha + \gamma \ | \gamma < \beta \}$ is a set. But in other order one doesn't know if the sum is uniquely defined.
We know $\{ \alpha + \gamma \ | \gamma < \beta \}$ is a set by the axiom of replacement. We replace each element of $\gamma$ by its addition with $\alpha$. For each element of $\gamma$ we have already defined that sum.