3. 모임과 집합 (iii): 첨수족과 멱집합Classes and Sets (iii): Indexed Families and Power Sets

1. Suppose that $A_i \subset B$ for all $i \in I$ and let $x \in \bigcup_{i \in I} A_i$. Then there exists a $j \in I$ such that $x \in A_j$. Since $A_j \subset B$, we have
   $$x \in B.$$
2. Suppose that $B \subset A_i$ for all $i \in I$ and let $x \in B$. Then $x \in A_i$ for all $i \in I$. Finally, by the definition of the intersection of the classes $A_i$, we have
   $$x \in \bigcap_{i \in I} A_i.$$

1. $$\begin{aligned} x \in \left( \bigcup_{i \in I} A_i \right)^c &\iff x \notin \bigcup_{i \in I} A_i \\ &\iff \text{there is no } j \in I \text{ such that } x \in A_j \\ &\iff \forall i \in I, x \notin A_i \\ &\iff \forall i \in I, x \in A_i^c \\ &\iff x \in \bigcap_{i \in I} A_i^c. \end{aligned}$$
2. $$\begin{aligned} x \in \left( \bigcap_{i \in I} A_i \right)^c &\iff x \notin \bigcap_{i \in I} A_i \\ &\iff \text{there exists a } j \in I \text{ such that } x \notin A_j \\ &\iff \exists j \in I \ni x \in A_j^c \\ &\iff x \in \bigcup_{i \in I} A_i^c. \end{aligned}$$

1. $$\begin{aligned} x \in \left( \bigcup_{i \in I} A_i \right) \cap \left( \bigcup_{j \in J} B_j \right) &\iff x \in \bigcup_{i \in I} A_i \quad \text{and} \quad x \in \bigcup_{j \in J} B_j \\ &\iff \exists\, h \in I \ni x \in A_h \quad \text{and} \quad \exists\, k \in J \ni x \in B_k \\ &\iff \exists\, h \in I \text{ and } \exists k \in J \ni x \in A_h \text{ and } x \in B_k \\ &\iff \exists\, (h, k) \in I \times J \ni x \in A_h \cap B_k \\ &\iff x \in \bigcup_{(i,j) \in I \times J} (A_i \cap B_j). \end{aligned}$$
2. $$\begin{aligned} x \in \left( \bigcap_{i \in I} A_i \right) \cup \left( \bigcap_{j \in J} B_j \right) &\iff x \in \left( \bigcap_{i \in I} A_i \right) \text{ or } \left( \bigcap_{j \in J} B_j \right) \\ &\iff x \in A_i \ \forall i \in I \text{ or } x \in B_j \ \forall j \in J \\ &\iff x \in A_i \text{ or } x \in B_j \quad \forall (i, j) \in I \times J \\ &\iff x \in A_i \cup B_j \quad \forall (i, j) \in I \times J \\ &\iff \bigcap_{(i,j) \in I \times J} (A_i \cup B_j). \end{aligned}$$

1. $$\begin{aligned} x \in dom \left( \bigcup_{i \in I} G_i \right) &\iff \exists y \ni (x, y) \in \left( \bigcup_{i \in I} G_i \right) \\ &\iff \exists y \ni (\exists j \in I \ni (x, y) \in G_j) \\ &\iff \exists\, (y \text{ and } j \in I) \ni (x, y) \in G_j \\ &\iff \exists j \in I \ni (\exists y \ni (x, y) \in G_j) \\ &\iff \exists j \in I \ni (x \in dom \ G_j) \\ &\iff x \in \bigcup_{i \in I} (dom \ G_i). \end{aligned}$$
2. $$\begin{aligned} y \in ran \left( \bigcup_{i \in I} G_i \right) &\iff \exists x \ni (x, y) \in \bigcup_{i \in I} G_i \\ &\iff \exists x \ni (\exists j \in I \ni (x, y) \in G_j) \\ &\iff \exists j \in I \ni (\exists x \ni (x, y) \in G_j) \\ &\iff \exists j \in I \ni y \in ran \ G_j \\ &\iff y \in \bigcup_{i \in I} (ran \ G_i). \end{aligned}$$

Recall that

and

We claim

Let $(x, y) \in A \times B$. Since $\{x\} \subset A \subset A \cup B$ and $\{x, y\} \subset A \cup B$, by the definition of the power set of $A \cup B$, $\{x\} \in \mathcal{P}(A \cup B)$ and $\{x, y\} \in \mathcal{P}(A \cup B)$. Thus

Moreover by the definition of the power set, we have

Therefore, the claim is proved. Note that by the axiom of union and the axiom of power set, $\mathcal{P} \left[ \mathcal{P}(A \cup B) \right]$ is a set. Finally by the axiom of subset and (3.17), $A \times B$ is a set

By Theorem 3.86, $A \times B$ is a set. Thus by the axiom of union, $\bigcup (A \times B)$ is a set.

Since $(a_1, b_1) \in A \times B$ and $(a_2, b_2) \in A \times B$,

Recall that $\bigcup (A \times B)$ is a set by Corollary 3.87. Thus by the axiom of subset, $(a_1, b_1) \cup (a_2, b_2)$ is a set. Moreover, by the definition of the ordered pair,

Finally, by the definition of union and the axiom of extent, we have

is a set.