The proof is by induction. Consider a single set $A_1$ Then the principle of inclusion-exclusion states that $|A_1| = |A_1|$ which is trivially true.

Now consider a collection of exactly two sets $A_1$ and $A_2$ We know that $$A \cup B = (A \setminus B) \cup (B \setminus A) \cup (A \cap B)$$ Furthermore, the three sets on the right-hand side of that equation must be disjoint. Therefore, by the addition principle, we have \begin{eqnarray*} |A \cup B| &=& |A \setminus B| + |B \setminus A| + |A \cap B| \\ &=& |A \setminus B | + |A \cap B| + |B \setminus A| + |A \cap B| - |A \cap B| \\ &=& |A| + |B| - |A \cap B| \\ \end{eqnarray*} So the principle of inclusion-exclusion holds for any two sets.

Now consider a collection of $N > 2$ finite sets $A_1, A_2, \dots A_N$ We assume that the principle of inclusion-exclusion holds for any collection of $M$ sets where $1 \leq M < N$ Because the union of sets is associative, we may break up the union of all sets in the collection into a union of two sets: $$\bigcup_{i=1}^N A_i = \left( \bigcup_{i=1}^{N-1} A_i \right ) \cup A_N$$

By the principle of inclusion-exclusion for two sets, we have $$\left | \bigcup_{i=1}^N A_i \right | = \left | \bigcup_{i=1}^{N-1} A_i \right| + |A_N| - \left|\left( \bigcup_{i=1}^{N-1} A_i \right ) \cap A_N \right|$$

Now, let $I_k$ be the collection of all $k$ fold intersections of $A_1, A_2, \dots A_{N-1}$ and let $I'_k$ be the collection of all $k$ fold intersections of $A_1, A_2, \dots A_N$ that include $A_N$ Note that $A_N$ is included in every member of $I'_k$ and in no member of $I_k$ so the two sets do not duplicate one another.

We then have $$\left | \bigcup_{i=1}^N A_i \right | = \sum_{j=1}^N \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + |A_N| - \left|\left( \bigcup_{i=1}^{N-1} A_i \right ) \cap A_N \right|$$ by the principle of inclusion-exclusion for a collection of $N-1$ sets. Then, we may distribute set intersection over set union to find that $$\left | \bigcup_{i=1}^N A_i \right | = \sum_{j=1}^N \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + |A_N| - \left| \bigcup_{i=1}^{N-1} (A_i \cap A_N) \right |$$ Note, however, that $$(A_x \cap A_N) \cup (A_y \cap A_N) = (A_x \cap A_y \cap A_N)$$ Hence we may again apply the principle of inclusion-exclusion for $N-1$ sets, revealing that \begin{eqnarray*} \left | \bigcup_{i=1}^N A_i \right | &=& \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + |A_N| - \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I_j} |S \cap A_N| \right ) \\ &=& \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + |A_N| - \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I'_{j+1}} |S| \right ) \\ &=& \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + |A_N| - \sum_{j=2}^{N} \left( (-1)^{(j)} \sum_{S\in I'_{j}} |S| \right ) \\ &=& \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + |A_N| + \sum_{j=2}^{N} \left( (-1)^{(j+1)} \sum_{S\in I'_{j}} |S| \right ) \\ \end{eqnarray*} The second sum does not include $I'_1$ Note, however, that $I'_1 = \{ A_N \}$ so we have \begin{eqnarray*} \left | \bigcup_{i=1}^N A_i \right | &=& \sum_{j=1}^{N-1} \left( (-1)^{(j+1)} \sum_{S\in I_j} |S| \right ) + \sum_{j=1}^{N} \left( (-1)^{(j+1)} \sum_{S\in I'_{j}} |S| \right ) \\ &=& \sum_{j=1}^{N-1} \left[ (-1)^{(j+1)} \left( \sum_{S\in I_j} |S| + \sum_{S\in I'_{j}} |S| \right ) \right] +(-1)^{N+1}\left | \bigcap_{i=1}^N A_i\right | \\ \end{eqnarray*} Combining the two sums yields the principle of inclusion-exclusion for $N$ sets.