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 ∩ { 𝐴 } ) = ∅

Proof

Step	Hyp	Ref	Expression
1		elirr	⊢ ¬ 𝐴 ∈ 𝐴
2		disjsn	⊢ ( ( 𝐴 ∩ { 𝐴 } ) = ∅ ↔ ¬ 𝐴 ∈ 𝐴 )
3	1 2	mpbir	⊢ ( 𝐴 ∩ { 𝐴 } ) = ∅