Let $\pi: A \to B(\mathcal{H})$ be a representation where $A$ is a $C^*$-algebra and $\mathcal{H}$ a Hilbert space. I'm trying to show the following four statements are equivalent:
(1) $\pi$ is irreducible.
(2) $\pi(A)' = \mathbb{C}1_{\mathcal{H}}$.
(3) $\pi(A)'$ contains no non-trivial projections.
(4) Every $h \in \mathcal{H}\setminus \{0\}$ is a cyclic vector for $\pi$.
I managed to show $(1) \implies (4)$ and $(2) \implies (3)$ and $(1) \implies (2)$ but I'm stuck on the other implications. I read other answers but most of these use Von Neumann Algebra theory, from which I know nothing, so please avoid using Von Neumann algebra facts to answer this question.
For the missing implication (3) $\Rightarrow$ (4), prove the counter-reciprocal. If there is a non-zero vector $h \in H$ such that it is not cyclic, then $\pi(A)'$ has a nontrivial projection. Let $K = \overline{\pi(A)h}$ be the cyclic subspace generated by $h$. It is non-null since $h \neq 0$ and standard approximate unit arguments and it is not $H$. Elementary calculations give that the projection $P_K \in B(H)$ belongs to $\pi(A)'$.