diff --git a/src/librustc_typeck/check/op.rs b/src/librustc_typeck/check/op.rs
index d2fe099e433f2..46c81d34d1b67 100644
--- a/src/librustc_typeck/check/op.rs
+++ b/src/librustc_typeck/check/op.rs
@@ -332,8 +332,9 @@ impl<'a, 'gcx, 'tcx> FnCtxt<'a, 'gcx, 'tcx> {
                                 op.node.as_str(),
                                 lhs_ty);
 
+                            let mut involves_fn = false;
                             if !lhs_expr.span.eq(&rhs_expr.span) {
-                                self.add_type_neq_err_label(
+                                involves_fn |= self.add_type_neq_err_label(
                                     &mut err,
                                     lhs_expr.span,
                                     lhs_ty,
@@ -341,7 +342,7 @@ impl<'a, 'gcx, 'tcx> FnCtxt<'a, 'gcx, 'tcx> {
                                     op,
                                     is_assign
                                 );
-                                self.add_type_neq_err_label(
+                                involves_fn |= self.add_type_neq_err_label(
                                     &mut err,
                                     rhs_expr.span,
                                     rhs_ty,
@@ -410,7 +411,7 @@ impl<'a, 'gcx, 'tcx> FnCtxt<'a, 'gcx, 'tcx> {
                                         "`{}` might need a bound for `{}`",
                                         lhs_ty, missing_trait
                                     ));
-                                } else if !suggested_deref {
+                                } else if !suggested_deref && !involves_fn {
                                     err.note(&format!(
                                         "an implementation of `{}` might \
                                          be missing for `{}`",
@@ -437,7 +438,7 @@ impl<'a, 'gcx, 'tcx> FnCtxt<'a, 'gcx, 'tcx> {
         other_ty: Ty<'tcx>,
         op: hir::BinOp,
         is_assign: IsAssign,
-    ) {
+    ) -> bool {
         err.span_label(span, ty.to_string());
         if let FnDef(def_id, _) = ty.sty {
             let source_map = self.tcx.sess.source_map();
@@ -481,8 +482,10 @@ impl<'a, 'gcx, 'tcx> FnCtxt<'a, 'gcx, 'tcx> {
                     variable_snippet,
                     applicability,
                 );
+                return true;
             }
         }
+        false
     }
 
     fn check_str_addition(
diff --git a/src/test/ui/fn/fn-compare-mismatch.stderr b/src/test/ui/fn/fn-compare-mismatch.stderr
index 74fb00f8ac383..c64070a2022d9 100644
--- a/src/test/ui/fn/fn-compare-mismatch.stderr
+++ b/src/test/ui/fn/fn-compare-mismatch.stderr
@@ -5,8 +5,6 @@ LL |     let x = f == g;
    |             - ^^ - fn() {main::g}
    |             |
    |             fn() {main::f}
-   |
-   = note: an implementation of `std::cmp::PartialEq` might be missing for `fn() {main::f}`
 help: you might have forgotten to call this function
    |
 LL |     let x = f() == g;
diff --git a/src/test/ui/issues/issue-59488.rs b/src/test/ui/issues/issue-59488.rs
index 27cf16a821ffc..d5fab0a6f9b26 100644
--- a/src/test/ui/issues/issue-59488.rs
+++ b/src/test/ui/issues/issue-59488.rs
@@ -10,17 +10,17 @@ fn bar(a: i64) -> i64 {
 
 fn main() {
     foo > 12;
-    //~^ ERROR 12:9: 12:10: binary operation `>` cannot be applied to type `fn() -> i32 {foo}` [E0369]
-    //~| ERROR 12:11: 12:13: mismatched types [E0308]
+    //~^ ERROR binary operation `>` cannot be applied to type `fn() -> i32 {foo}` [E0369]
+    //~| ERROR mismatched types [E0308]
 
     bar > 13;
-    //~^ ERROR 16:9: 16:10: binary operation `>` cannot be applied to type `fn(i64) -> i64 {bar}` [E0369]
-    //~| ERROR 16:11: 16:13: mismatched types [E0308]
+    //~^ ERROR binary operation `>` cannot be applied to type `fn(i64) -> i64 {bar}` [E0369]
+    //~| ERROR mismatched types [E0308]
 
     foo > foo;
-    //~^ ERROR 20:9: 20:10: binary operation `>` cannot be applied to type `fn() -> i32 {foo}` [E0369]
+    //~^ ERROR binary operation `>` cannot be applied to type `fn() -> i32 {foo}` [E0369]
 
     foo > bar;
-    //~^ ERROR 23:9: 23:10: binary operation `>` cannot be applied to type `fn() -> i32 {foo}` [E0369]
-    //~| ERROR 23:11: 23:14: mismatched types [E0308]
+    //~^ ERROR binary operation `>` cannot be applied to type `fn() -> i32 {foo}` [E0369]
+    //~| ERROR mismatched types [E0308]
 }
diff --git a/src/test/ui/issues/issue-59488.stderr b/src/test/ui/issues/issue-59488.stderr
index b49f5e35f4226..615adb36d330c 100644
--- a/src/test/ui/issues/issue-59488.stderr
+++ b/src/test/ui/issues/issue-59488.stderr
@@ -6,8 +6,6 @@ LL |     foo > 12;
    |     |
    |     fn() -> i32 {foo}
    |     help: you might have forgotten to call this function: `foo()`
-   |
-   = note: an implementation of `std::cmp::PartialOrd` might be missing for `fn() -> i32 {foo}`
 
 error[E0308]: mismatched types
   --> $DIR/issue-59488.rs:12:11
@@ -26,8 +24,6 @@ LL |     bar > 13;
    |     |
    |     fn(i64) -> i64 {bar}
    |     help: you might have forgotten to call this function: `bar( /* arguments */ )`
-   |
-   = note: an implementation of `std::cmp::PartialOrd` might be missing for `fn(i64) -> i64 {bar}`
 
 error[E0308]: mismatched types
   --> $DIR/issue-59488.rs:16:11
@@ -45,8 +41,6 @@ LL |     foo > foo;
    |     --- ^ --- fn() -> i32 {foo}
    |     |
    |     fn() -> i32 {foo}
-   |
-   = note: an implementation of `std::cmp::PartialOrd` might be missing for `fn() -> i32 {foo}`
 help: you might have forgotten to call this function
    |
 LL |     foo() > foo;