Using Farkas Lemma to deduce $0^Tx<-1$

Question

Given the polyhedron $P=\{x \in \mathbb{R}^n\ : Ax\leq b\}$ for $A \in \mathbb{R}^{m\times n}$, $b \in \mathbb{R}^m$ and a vector $w \in \mathbb{Z}^n$. Now consider $\hat{P} = \{x \in P : w^Tx \leq \lfloor x \rfloor \}$. If we know that $\hat{P} = \emptyset$, then why can we deduce with Farkas' Lemma that $0^Tx\leq -1$?