Exponential of Real Number is Strictly Positive/Proof 5

Theorem
Let $x$ be a real number.

Let $\exp$ denote the (real) Exponential Function.

Then:
 * $\forall x \in \R: \exp x > 0$

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 \exp \left({0}\right) = 1$

on $\R$.

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

Then $0 \in \left({\exp \alpha \,.\,.\, 1}\right)$.

From Intermediate Value Theorem:
 * $\exists \zeta \in \left({\alpha \,.\,.\, 0}\right): f \left({\zeta}\right) = 0$

This contradicts the lemma.