Showing a Set is Uncountable (Using Cantor's Diagonalization)

Proof 1 (using cardinality)

Let $$\begin{array}{l|rcl} \psi : & \mathcal P(\mathbb N) & \longrightarrow & L \\ & k & \longmapsto & \{\langle 1, l\rangle \mid l \in k\} \end{array}$$ $\psi$ is injective. As $\vert \mathcal P(\mathbb N) \vert \le \vert L \vert \le \vert \mathcal P(\mathbb N \times \mathbb N) \vert = \mathfrak c \gt \aleph_0$, where $\mathfrak c$ stands for the cardinality of the continuum, we get the desired result.

Proof 2 (diagonal argument)

Suppose that $\varphi: \mathbb{N} \rightarrow L$ is a bijection. Let $l \in \mathcal P(\mathbb N \times \mathbb N)$ be defined as

$$l = \{\langle 1, m\rangle \mid m \notin p_2(\varphi(m))\}$$ where $p_2(u) = \{u_2 \mid \langle u_1, u_2 \rangle \in u\}$ for any $u \in \mathcal P(\mathbb N \times \mathbb N)$. We have $l \in L$ and get a contradiction if we raise the question $l \in \varphi[\mathbb N]$?