diff --git a/compiler/rustc_hir_analysis/src/coherence/mod.rs b/compiler/rustc_hir_analysis/src/coherence/mod.rs
index fc8fab0eabc83..1f73950765ed0 100644
--- a/compiler/rustc_hir_analysis/src/coherence/mod.rs
+++ b/compiler/rustc_hir_analysis/src/coherence/mod.rs
@@ -43,7 +43,7 @@ fn enforce_trait_manually_implementable(
     let impl_header_span = tcx.def_span(impl_def_id);
 
     // Disallow *all* explicit impls of traits marked `#[rustc_deny_explicit_impl]`
-    if tcx.trait_def(trait_def_id).deny_explicit_impl {
+    if tcx.trait_def(trait_def_id).deny_explicit_impl && !tcx.hir().rustc_coherence_is_core() {
         let trait_name = tcx.item_name(trait_def_id);
         let mut err = struct_span_err!(
             tcx.sess,
diff --git a/library/core/src/any.rs b/library/core/src/any.rs
index 8f5404d9713dd..5e4346c16c9c0 100644
--- a/library/core/src/any.rs
+++ b/library/core/src/any.rs
@@ -112,6 +112,7 @@ use crate::intrinsics;
 // but we would likely want to indicate as such in documentation).
 #[stable(feature = "rust1", since = "1.0.0")]
 #[cfg_attr(not(test), rustc_diagnostic_item = "Any")]
+#[cfg_attr(not(bootstrap), rustc_deny_explicit_impl(implement_via_object = true))]
 pub trait Any: 'static {
     /// Gets the `TypeId` of `self`.
     ///
@@ -132,7 +133,7 @@ pub trait Any: 'static {
 }
 
 #[stable(feature = "rust1", since = "1.0.0")]
-impl<T: 'static + ?Sized> Any for T {
+impl<T: 'static> Any for T {
     fn type_id(&self) -> TypeId {
         TypeId::of::<T>()
     }