# How can I make Mathematica print the intersection of two intervals?

I'm trying to use NumberLinePlot and Manipulate to visualize how the interval $$[t-1, t] \cap [0, 1]$$ changes as the value of $$t$$ changes. My code is:

Manipulate[NumberLinePlot[{Interval[{0, 1}], Interval[{t - 1, t}]}, PlotRange -> {-3, 3}], {t, -3, 3}]


which gives a result that looks like this: My question is: how can I make Mathematica tell me explicitly what the intersection is? For example, with the slider positioned as in the picture, I would want it to print something like $$[0, 0.6]$$

Manipulate[
Module[{int1, int2},
int1 = Interval[{0, 1}];
int2 = Interval[{t - 1, t}];

Grid[{{IntervalIntersection[int1, int2]},
{NumberLinePlot[{int1, int2}, PlotRange -> {-3, 3}]}
}]
],
{t, -3, 3}, TrackedSymbols :> {t}]


Some embellishments on answer by @Nasser

Clear["Global*"]

Manipulate[Module[{int}, int[1] = Interval[{0, 1}];
int[2] = Interval[{t - 1, t}];
int[3] = IntervalIntersection[
int[1], int[2]];
Grid[{
{If[int[3] === Interval[],
Nothing, int[3]]}, {NumberLinePlot[int /@ {1, 2, 3},
PlotRange -> {-2, 3}]}}]], {{t, 0.5}, -1, 3, 0.01,
Appearance -> "Labeled"}, TrackedSymbols :> {t}]
`