Let $X : \Omega \to \mathcal{X}$ be a D5723: Simple random variable.
Let $a \in (0, \infty) \setminus \{ 1 \}$ be a D5407: Positive real number.

Using results

(i)	R4847: Probability of event conditional on itself
(ii)	R4828: Logarithm of one is zero

\begin{equation} \begin{split} H_a(X \mid X) & = - \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} \mathbb{P}(X = x, X = y) \log_a \mathbb{P}(X = x \mid X = y) \\ & = - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x, X = x) \log_a \mathbb{P}(X = x \mid X = x) \\ & = - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a 1 \\ & = - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \cdot 0 \\ \end{split} \end{equation} $\square$