Proving ∃x(P(x) → ∀y(P(y))): An Exercise in Logic

[Syrus]

Homework Statement

Prove ∃x(P(x) → ∀y(P(y))).

Homework Equations

The Attempt at a Solution

∃x(P(x) → ∀y(P(y))) is equivalent to ∃x(¬P(x) ∨ ∀y(P(y))).



This exercise is found in a section on "proofs involving disjunctions." I have tried many different ways to solve this and have a feeling I am not approaching it the right way. Perhaps I am not considering clever enough exhaustive cases? Can anyone direct me in the proper way to begin?

 

[Dansuer]

This is an exercise from "How to prove it" right?

I'm having troubles with this one too.

I've tried a proof by contradiction, i think i proved it but I'm not so sure.