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aa = ContourPlot[
   Sin[x - y] + 
    Cos[x + y],(*this function is just for the MWE*){x, -10, 
    10}, {y, -10, 10}, PlotPoints -> {30, 30}, 
   PlotRange -> {{-10, 10}, {-10, 10}}, ColorFunction -> "Rainbow", 
   ImageSize -> 500, MeshFunctions -> (#3 &), Mesh -> None, 
   ContourStyle -> None, ContourShading -> Automatic];
bb = ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
   PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
   ImageSize -> 500, ContourShading -> None, 
   Contours -> {{-1, {Thick, Red}}, {0, {Thick, Green}}, {1, {Thick, 
       Blue}}}];
Show[aa, bb]

aa = ContourPlot[
   Sin[x - y] + 
    Cos[x + y],(*this function is just for the MWE*){x, -10, 
    10}, {y, -10, 10}, PlotPoints -> {30, 30}, 
   PlotRange -> {{-10, 10}, {-10, 10}}, ColorFunction -> "Rainbow", 
   ImageSize -> 500];
bb = ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
   PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
   ImageSize -> 500, ContourShading -> None, 
   Contours -> {{-1, {Thick, Red}}, {0, {Thick, Green}}, {1, {Thick, 
       Blue}}}];
Show[aa, bb]

Or this?

aa = ContourPlot[
   Sin[x - y] + 
    Cos[x + y],(*this function is just for the MWE*){x, -10, 
    10}, {y, -10, 10}, PlotPoints -> {30, 30}, 
   PlotRange -> {{-10, 10}, {-10, 10}}, ColorFunction -> "Rainbow", 
   ImageSize -> 500, MeshFunctions -> (#3 &), Mesh -> None, 
   ContourStyle -> None, ContourShading -> Automatic];
bb = ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
   PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
   ImageSize -> 500, ContourShading -> None, ContourStyle -> None, 
   MeshFunctions -> (-#3 &), 
   Mesh -> {{{-1, {Thick, Red, Opacity[1]}}, {0, {Thick, Green, 
        Opacity[1]}}, {1, {Thick, Blue, Opacity[1]}}}}];
Show[aa, bb]

2D

aa = ContourPlot[
   Sin[x - y] + 
    Cos[x + y],(*this function is just for the MWE*){x, -10, 
    10}, {y, -10, 10}, PlotPoints -> {30, 30}, 
   PlotRange -> {{-10, 10}, {-10, 10}}, ColorFunction -> "Rainbow", 
   ImageSize -> 500, MeshFunctions -> (#3 &), Mesh -> None, 
   ContourStyle -> None, ContourShading -> Automatic];
bb = ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
   PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
   ImageSize -> 500, MeshFunctions -> {-#3 &}, 
   Mesh -> {{{-1, {Thick, Red, Opacity[1]}}, {0, {Thick, Green, 
        Opacity[1]}}, {1, {Thick, Blue, Opacity[1]}}}}, 
   ContourStyle -> None, ContourShading -> None];
Show[aa, bb]

2D

Set Contours

Contours -> {{-1, {Thick, Red}}, {0, {Thick, Green}}, {1, {Thick, 
       Blue}}}

aa = ContourPlot[
   Sin[x - y] + 
    Cos[x + y],(*this function is just for the MWE*){x, -10, 
    10}, {y, -10, 10}, PlotPoints -> {30, 30}, 
   PlotRange -> {{-10, 10}, {-10, 10}}, ColorFunction -> "Rainbow", 
   ImageSize -> 500, MeshFunctions -> (#3 &), Mesh -> None, 
   ContourStyle -> None, ContourShading -> Automatic];
bb = ContourPlot[Sin[x - y] + Cos[x + y], {x, -10, 10}, {y, -10, 10}, 
   PlotPoints -> {30, 30}, PlotRange -> {{-10, 10}, {-10, 10}}, 
   ImageSize -> 500, ContourShading -> None, 
   Contours -> {{-1, {Thick, Red}}, {0, {Thick, Green}}, {1, {Thick, 
       Blue}}}];
Show[aa, bb]