Compare Two Lists Via One Property Using LINQ

You can do this

widgets2.Where(y=>widget1.Any(z=>z.TypeID==y.TypeID));

Join has the drawback that your results might be duplicated if widgets1 or widgets2 contains elements with the same TypeID more than one (which also applies to your original code, by the way).

The following will do exactly what you want: Return all elements from widgets1 for which an element with a corresponding TypeID exists in widgets2.

widgets1_in_widgets2 = (from w1 in widgets1
                        where widgets2.Any(w2 => w1.TypeID == w2.TypeID)
                        select w1).ToList()

I like this solution because it is simple to read in the code.

bool result = firstList.All(o => secondList.Any(w => w.Prop1 == o.Prop1 && w.Prop2 == o.Prop2));

See the full example in fiddle: Fiddle example comparation

What you want here is a Join.

var widgets1_in_widgets2 = from first in widgest1
    join second in widgets2
    on first.TypeID equals second.TypeID
    select first;

Intersect can be more or less thought of as a special case of Join where the two sequences are of the same type, and can thus be applied for equality instead of needing a projection for each type to generate a key to compare. Given your case, Intersect isn't an option.

If a particular ID is duplicated in your second set and you don't want the item to be duplicated in the results then you can use a GroupJoin instead of a Join:

var widgets1_in_widgets2 = from first in widgest1
    join second in widgets2
    on first.TypeID equals second.TypeID
    into matches
    where matches.Any()
    select first;