Exponential of Real Number is Strictly Positive/Proof 5

Proof
This proof assumes the definition of $\exp$ as the solution to an initial value problem.

That is, suppose $\exp$ satisfies:
 * $ (1): \quad D_x \exp x = \exp x$
 * $ (2): \quad \map \exp 0 = 1$

on $\R$.

Lemma
that $\exists \alpha \in \R: \exp \alpha < 0$.

Then $0 \in \openint {\exp \alpha} 1$.

From Intermediate Value Theorem:
 * $\exists \zeta \in \openint \alpha 0: \map f \zeta = 0$

This contradicts the lemma.