diff --git a/Lib/_abcoll.py b/Lib/_abcoll.py
index cac06e0..0957553 100644
--- a/Lib/_abcoll.py
+++ b/Lib/_abcoll.py
@@ -90,7 +90,8 @@
     @classmethod
     def __subclasshook__(cls, C):
         if cls is Iterator:
-            if any("__next__" in B.__dict__ for B in C.__mro__):
+            if (any("__next__" in B.__dict__ for B in C.__mro__) and
+                any("__iter__" in B.__dict__ for B in C.__mro__)):
                 return True
         return NotImplemented
 
diff --git a/Lib/test/test_collections.py b/Lib/test/test_collections.py
index f3f8b03..02b9dc3 100644
--- a/Lib/test/test_collections.py
+++ b/Lib/test/test_collections.py
@@ -356,8 +356,14 @@
         for x in samples:
             self.assertIsInstance(x, Iterator)
             self.assertTrue(issubclass(type(x), Iterator), repr(type(x)))
-        self.validate_abstract_methods(Iterator, '__next__')
-        self.validate_isinstance(Iterator, '__next__')
+        self.validate_abstract_methods(Iterator, '__next__', '__iter__')
+
+        # Issue 10565
+        class NextOnly:
+            def __next__(self):
+                yield 1
+                raise StopIteration
+        self.assertNotIsInstance(NextOnly(), Iterator)
 
     def test_Sized(self):
         non_samples = [None, 42, 3.14, 1j,