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 e. V -> ( A e. { B } <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		elsni	|- ( A e. { B } -> A = B )
2		snidg	|- ( B e. V -> B e. { B } )
3		eleq1	|- ( A = B -> ( A e. { B } <-> B e. { B } ) )
4	2 3	syl5ibrcom	|- ( B e. V -> ( A = B -> A e. { B } ) )
5	1 4	impbid2	|- ( B e. V -> ( A e. { B } <-> A = B ) )