I'm studying analysis 1 and I have a question.
Let $X$ and $Y$ be finite sets. Then $X \cup Y$ is finite and $\#(X \cup Y) \le \# X + \# Y $. ($\#X$ means the cardinality of $X$)
This is the statement that I proved and I proved it by using mathematical induction just because I felt like I should use it.
And I know that mathematical induction skill is used for proving some statement like this form "For any natural number~,"
But if I could see the above bold statement as a conditional statement, then I think the antecedent would be "Let $X$ and $Y$ be finite sets." and the consequent would be "Then $X\cup Y$ is finite and $\#(X\cup Y) \le \#X+\#Y$."
So I thought that the first order logic expression for the antecedent would be like this "$\exists n\, \exists m\, \#X=n,\ \#Y=m$" because of the definition of a finite set(a set is finite iff it has cardinality $n$ for some natural number) and the whole expression would be "$\exists n\ \exists m\, \#X=n,\ \#Y=m \rightarrow \exists a\, \#(X\cup Y)=a\land a\le n+m$"
But if it's expressed like this, there's no reason to use mathematical induction because there is no expression $\forall$ in that formula.
What is the problem?