Given the following integer linear program,
(ILP) $\min c'x$ subject to $Ax \ge b, x \in \mathbb{N}_0$
where all elements of $A$ and $b$ are integral, and assuming its linear-program relaxation (LP) has a non-degenerate optimal solution $x^\mathrm{LP}$ with at least one fractional valued variable, we can formulate the Dantzig cut $$\sum_{i\in N}x_i \ge 1$$ where $N$ is the set of nonbasic variables at $x^\mathrm{LP}$. The reason is, since $x^\mathrm{LP}$ is not feasible in (ILP), any optimal solution of (ILP) cannot have all $x_i, x\in N,$ zero; i.e., at least one of $x_i, i\in N$, is strictly positive.
Here is my question: Where does the condition that $A$ and $b$ are integral come in? It seems to me that every step of the argument goes through without this assumption.