Theorem sneqbg

Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012)

	Ref	Expression
Assertion	sneqbg	$$⊢A\in V\to \left(\left\{A\right\}=\left\{B\right\}\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sneqrg	$$⊢A\in V\to \left(\left\{A\right\}=\left\{B\right\}\to A=B\right)$$
2		sneq	$$⊢A=B\to \left\{A\right\}=\left\{B\right\}$$
3	1 2	impbid1	$$⊢A\in V\to \left(\left\{A\right\}=\left\{B\right\}\leftrightarrow A=B\right)$$