I want to prove that set $A$ is empty ($A = \varnothing$).
Intuitively, I understand that a way to do it would to be to falsely assume $ x \in A$ and show a contradiction (because if it's empty, $x \notin A$).
But I don't understand how that works mathematically.
Mathematically, I'm tripping because I need to prove is $A \subseteq \varnothing$, which would mean that $\forall x. \text{ if } x \in A \text{ then } x \in \varnothing$ (I'm aware that doesn't makes sense, but still required, which is why I'm confused).
According to this question, the contradiction really is enough, but why? how does that help you prove $A \subseteq \varnothing$, for you to be ultimately able to say $A=\varnothing$?
Best Answer
Because $x \in \varnothing$ is identically false, the only way $x \in A \implies x \in \varnothing$ can be true is if $x \in A$ is false.
Thus, the only way $\forall x: x \in A \implies x \in \varnothing$ can be true is if $x \in A$ is false for all $x$.