Coq 如何证明两个不相等单体集的交集为空

我把证明的内容归结为两个不同的单态集的交集是空的，但我无法证明这个看似简单的事实

我查阅了集合库、动力集事实、构造集等，但没有找到任何有用的东西

Require Import Coq.Sets.Ensembles.
Require Import Coq.Strings.String.

Example x: string := "x".
Example y: string := "y".

Lemma ex:
Intersection string (Singleton string x)
  (Singleton string y) = Empty_set string.
Proof.
  ???

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关键是使用可扩展性公理：

Require Import Coq.Sets.Ensembles.
Require Import Coq.Strings.String.

Example x: string := "x".
Example y: string := "y".

Lemma ex:
Intersection string (Singleton string x)
  (Singleton string y) = Empty_set string.
Proof.
apply Extensionality_Ensembles. split.
- intros _ [b Ha Hb].
  inversion Ha. inversion Hb. unfold x, y in *. congruence.
- now intros _ [].
Qed.