I need to prove that if $n \in \mathbb{N} $ and $A \subseteq I_n $, then $A$ is finite and $|A| \leq n$. Furthermore, if $A \ne I_n$, then $|A| < n$. Here,
$$ I_n = \{ i \in \mathbb{Z}^+ \mid i \leq n \} $$
Since there is a natural number involved here, its wise to use mathematical induction. I think I need two implication statements here. $P(n)$ and $Q(n)$. So, general form here would be to prove that
$$ \forall \, n \in \mathbb{N} \, \forall A \left[ P(n) \right] $$
$$ \forall \, n \in \mathbb{N} \, \forall A \left[ Q(n) \right] $$
Universe of discourse for $A$ would be all subsets of $I_n$. And I think $P(n)$ and $Q(n)$ will have to be some implication statement.
$$ P(n): \left[ A \subseteq I_n \Longrightarrow \text{ A is finite and }|A| \leq n \right]$$
$$ Q(n): \left[ A \ne I_n \Longrightarrow \text{ A is finite and }|A| < n \right]$$
And since universal quantifier distributes over conjunction, I can shorten it to be
$$ \forall \, n \in \mathbb{N} \, \forall A \left[ P(n) \,\wedge \, Q(n)\right] $$
Am I in the right direction ?
