diff --git a/NEWS.md b/NEWS.md
index b512e2c1b363b..429084fb4e451 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -362,9 +362,14 @@ This section lists changes that do not have deprecation warnings.
     trait; see its documentation for details. Types which support subtraction (operator
     `-`) must now implement `widen` for hashing to work inside heterogeneous arrays.
 
+  * `AbstractSet` objects are now considered equal by `==` and `isequal` if all of their
+    elements are equal ([#25368]). This has required changing the hashing algorithm
+    for `BitSet`.
+
   * `findn(x::AbstractVector)` now return a 1-tuple with the vector of indices, to be
     consistent with higher order arrays ([#25365]).
 
+
 Library improvements
 --------------------
 
diff --git a/base/bitset.jl b/base/bitset.jl
index 1fe60771a13b7..9f9f4f1dbb94b 100644
--- a/base/bitset.jl
+++ b/base/bitset.jl
@@ -326,7 +326,7 @@ function _check0(a::Vector{UInt64}, b::Int, e::Int)
     true
 end
 
-function issetequal(s1::BitSet, s2::BitSet)
+function ==(s1::BitSet, s2::BitSet)
     # Swap so s1 has always the smallest offset
     if s1.offset > s2.offset
         s1, s2 = s2, s1
@@ -356,7 +356,7 @@ function issetequal(s1::BitSet, s2::BitSet)
     return true
 end
 
-⊆(a::BitSet, b::BitSet) = a == intersect(a,b)
+issubset(a::BitSet, b::BitSet) = a == intersect(a,b)
 ⊊(a::BitSet, b::BitSet) = a <= b && a != b
 
 
diff --git a/base/set.jl b/base/set.jl
index 972ada8638149..c8cee2bdbe866 100644
--- a/base/set.jl
+++ b/base/set.jl
@@ -261,9 +261,9 @@ function symdiff!(s::AbstractSet, itr)
     s
 end
 
+==(l::AbstractSet, r::AbstractSet) = length(l) == length(r) && l ⊆ r
 # convenience functions for AbstractSet
-# (if needed, only their synonyms issetequal, ⊊ and ⊆ must be specialized)
-==(l::AbstractSet, r::AbstractSet) = issetequal(l, r)
+# (if needed, only their synonyms ⊊ and ⊆ must be specialized)
 <( l::AbstractSet, r::AbstractSet) = l ⊊ r
 <=(l::AbstractSet, r::AbstractSet) = l ⊆ r
 
@@ -284,7 +284,7 @@ julia> issubset([1, 2, 3], [1, 2])
 false
 ```
 """
-function ⊆(l, r)
+function issubset(l, r)
     for elt in l
         if !in(elt, r)
             return false
@@ -295,7 +295,7 @@ end
 # use the implementation below when it becoms as efficient
 # issubset(l, r) = all(_in(r), l)
 
-const issubset = ⊆
+const ⊆ = issubset
 
 """
     issetequal(a, b)
@@ -313,6 +313,7 @@ true
 ```
 """
 issetequal(l, r) = length(l) == length(r) && l ⊆ r
+issetequal(l::AbstractSet, r::AbstractSet) = l == r
 
 ⊊(l, r) = length(l) < length(r) && l ⊆ r
 ⊈(l, r) = !⊆(l, r)