Pauli exclusion principle

Two Fermions cannot be in the same Quantum state $|a⟩$.

Proof

Let $|a⟩$ be the Quantum state of a Fermion. Then

$$\begin{array}{rll}|\psi (a,a)⟩& =-|\psi (a,a)⟩& \text{Due to the anti-symmetry of fermions}\\ & \Downarrow & \\ |\psi (a,a)⟩& =0& \\ & \Downarrow & \\ \Vert \psi \Vert & \ne 1& \end{array}$$

Since $\psi$ is not Normalizable, it is not a valid Quantum state.