Let $X, Y \in \text{Random}(\mathcal{X})$ each be a D5723: Simple random variable.
Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.

Then

(1)	\begin{equation} - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(X = x) \leq - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(Y = x) \end{equation}
(2)	\begin{equation} - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(X = x) = - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(Y = x) \quad \iff \quad \forall \, x \in \mathcal{X} : \mathbb{P}(X = x) = \mathbb{P}(Y = x) \end{equation}