Theorem elsn2g

Description: There is exactly one element in a singleton. Exercise 2 of TakeutiZaring p. 15. This variation requires only that B , rather than A , be a set. (Contributed by NM, 28-Oct-2003)

	Ref	Expression
Assertion	elsn2g	$$⊢B\in V\to \left(A\in \left\{B\right\}\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		elsni	$$⊢A\in \left\{B\right\}\to A=B$$
2		snidg	$$⊢B\in V\to B\in \left\{B\right\}$$
3		eleq1	$$⊢A=B\to \left(A\in \left\{B\right\}\leftrightarrow B\in \left\{B\right\}\right)$$
4	2 3	syl5ibrcom	$$⊢B\in V\to \left(A=B\to A\in \left\{B\right\}\right)$$
5	1 4	impbid2	$$⊢B\in V\to \left(A\in \left\{B\right\}\leftrightarrow A=B\right)$$