# Intuition behind $\exists x \in S, \forall y \in \emptyset, P(x)$

I am trying to prove the statement in the form:

$$\exists x \in S, \forall y \in \emptyset, P(x)$$

where $$P(x): x > y$$ and $$S$$ is an arbitrary non-empty set.

My proof looks something like:

Let $$x =$$ some number in set S. Let $$y \in \emptyset$$. My intuition is that since we cannot check whether $$x > y$$ since $$y$$ is an empty set, the statement is vacuously true. My questions is, how do I write this "intuition" in a rigorous way?

• What you have written is already rigorous. – Kavi Rama Murthy 2 days ago

You could restate the claim as $$\exists x\in S\neg\exists y\in\emptyset\neg P(x)$$, which is trivial for each $$x\in S$$.
This reduces to: $$\exists x \colon y \in \{\} \implies P(x)$$.
But $$y \notin \{\}$$.
• I think you mean $\exists x,\,\forall y,\,(y\in\{\}\Rightarrow P(x))$ . – Gae. S. 2 days ago