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

Question

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 $[0, 0.6]$

Answer 1

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}]

Answer 2

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}]