Theorem disjcsn

Description: A class is disjoint from its singleton. A consequence of regularity. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Revised by BJ, 4-Apr-2019)

	Ref	Expression
Assertion	disjcsn	$$⊢A\cap \left\{A\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		elirr	$$⊢\neg A\in A$$
2		disjsn	$$⊢A\cap \left\{A\right\}=\varnothing \leftrightarrow \neg A\in A$$
3	1 2	mpbir	$$⊢A\cap \left\{A\right\}=\varnothing$$