Theorem elsng

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15 (generalized). (Contributed by NM, 13-Sep-1995) (Proof shortened by Andrew Salmon, 29-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		eqeq1	$$⊢x=y\to \left(x=B\leftrightarrow y=B\right)$$
2		eqeq1	$$⊢y=A\to \left(y=B\leftrightarrow A=B\right)$$
3		df-sn	$$⊢\left\{B\right\}=\left\{x|x=B\right\}$$
4	1 2 3	elab2gw	$$⊢A\in V\to \left(A\in \left\{B\right\}\leftrightarrow A=B\right)$$