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There is a huge recreation area but inside this recreation area there is some other zoning.

I finished creating all polygons but this recreation area polygon overlaps with the small zoning polygons within the same area.

What tool can I use to cut out the small zoning polygons from this big polygon?

I cannot use the erase tool as I don't have the license.

I am using ArcGIS Pro as well, which is really confusing since I just started using it.

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You could use the union tool with a single input layer. See more details about union here.

The union tool will create multiple output polygons where multiple input polygon intersect. You can select those polygons using select by location (have their centroid inside) after selecting by attributes those with "other zoning". In an edit session, you can then delete all the polygons selected (it is "in theory" possilbe but very very very unlikely that one or 2 big polygons are selected at this stage, so unselect them manually if necessary). You can also select further by attributes if you want to keep one of the the two island polygons.

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