Theorem snsssn

Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006)

	Ref	Expression
Hypothesis	sneqr.1	$$⊢A\in \mathrm{V}$$
Assertion	snsssn	$$⊢\left\{A\right\}\subseteq \left\{B\right\}\to A=B$$

Proof

Step	Hyp	Ref	Expression
1		sneqr.1	$$⊢A\in \mathrm{V}$$
2		sssn	$$⊢\left\{A\right\}\subseteq \left\{B\right\}\leftrightarrow \left(\left\{A\right\}=\varnothing \vee \left\{A\right\}=\left\{B\right\}\right)$$
3	1	snnz	$$⊢\left\{A\right\}\ne \varnothing$$
4	3	neii	$$⊢\neg \left\{A\right\}=\varnothing$$
5	4	pm2.21i	$$⊢\left\{A\right\}=\varnothing \to A=B$$
6	1	sneqr	$$⊢\left\{A\right\}=\left\{B\right\}\to A=B$$
7	5 6	jaoi	$$⊢\left(\left\{A\right\}=\varnothing \vee \left\{A\right\}=\left\{B\right\}\right)\to A=B$$
8	2 7	sylbi	$$⊢\left\{A\right\}\subseteq \left\{B\right\}\to A=B$$