Theorem disjsn2

Description: Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998)

	Ref	Expression
Assertion	disjsn2	$$⊢A\ne B\to \left\{A\right\}\cap \left\{B\right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		elsni	$$⊢B\in \left\{A\right\}\to B=A$$
2	1	eqcomd	$$⊢B\in \left\{A\right\}\to A=B$$
3	2	necon3ai	$$⊢A\ne B\to \neg B\in \left\{A\right\}$$
4		disjsn	$$⊢\left\{A\right\}\cap \left\{B\right\}=\varnothing \leftrightarrow \neg B\in \left\{A\right\}$$
5	3 4	sylibr	$$⊢A\ne B\to \left\{A\right\}\cap \left\{B\right\}=\varnothing$$