Here's a one-liner with another toolkit:

$ bedops -m A.bed B.bed | bedmap --echo-map --exact - <(bedops -u A.bed B.bed) | sort-bed - > answer.bed

Or if you only have one file:

$ bedops -m C.bed | bedmap --echo-map --exact - C.bed > answer.bed

It is assumed A.bed and B.bed and C.bed are sorted before set operations, i.e.:

$ sort-bed A.unsorted.bed > A.bed
$ sort-bed B.unsorted.bed > B.bed
$ sort-bed C.unsorted.bed > C.bed

Demo:

$ more C.bed
chr1    100     120     id-1
chr1    110     112     id-2
chr1    114     117     id-3
chr1    125     130     id-4

$ bedops -m C.bed | bedmap --echo-map --exact - C.bed
chr1    100 120 id-1
chr1    125 130 id-4

Another option is to filter the input file for purely nested elements.

Nested elements are a pain, but there are ways to deal with them.

Start with a utility awk script:

#!/usr/bin/env awk -f
{
    if (NR > 1) {
        currentChr = $1
        currentStart = $2
        currentStop = $3
        if ((previousStart < currentStart) && (previousStop > currentStop)) {
            print $0;
        }
        else {
            previousChr = currentChr
            previousStart = currentStart
            previousStop = currentStop
        }
    }
    else {
        previousChr = $1
        previousStart = $2
        previousStop = $3
    }
}

Let's call that script getNestedElements.awk. Make sure it is executable (chmod u+x ./getNestedElements.awk).

You can use this script to identify elements in A2.bed that are nested, and you would then pipe them to bedops --not-element-of 100% to filter them out:

$ more A2.bed
chr1    80      101     id-0
chr1    100     120     id-1
chr1    110     112     id-2
chr1    114     117     id-3
chr1    125     130     id-4

$ ./getNestedElements.awk A2.bed | bedops --not-element-of 100% A2.bed -
chr1    80      101     id-0
chr1    100     120     id-1
chr1    125     130     id-4