# $Q(n)$: "$P(k)$ holds for all $k<n$". Then why is $Q(0)$ clearly true? [duplicate]

It is a principle and proof from Introduction to Set Theory, Hrbacek and Jech.

In the proof, line 1 and 2, I couldn't understand why $$Q(0)$$ is true.

$$Q(0)$$ means that "$$P(k)$$ holds for all $$k<0$$".

I understood there are no $$k<0$$.

And then I couldn't proceed.

$$Q(n)$$ is the predicate "$$\forall k\in{\Bbb N}_0 (k." Then $$Q(0)$$ is vacuously true, since the premise is false and so the implication is true.
A sentence like "for all $$x$$, if $$x$$ has the property $$A$$, then $$x$$ has the property $$B$$" is false if (and only if) there exists a counterexample, that is, if there exists some $$x$$ with the property $$A$$ but without the property $$B$$.
If there is no $$x$$ that has the property $$A$$, then there is no counterexample, so the sentence is true.