Theorem snssg

Description: The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of Quine p. 49. (Contributed by NM, 22-Jul-2001) (Proof shortened by BJ, 1-Jan-2025)

	Ref	Expression
Assertion	snssg	|- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Proof

Step	Hyp	Ref	Expression
1		snssb	|- ( { A } C_ B <-> ( A e. _V -> A e. B ) )
2	1	bicomi	|- ( ( A e. _V -> A e. B ) <-> { A } C_ B )
3		elex	|- ( A e. V -> A e. _V )
4		imbibi	|- ( ( ( A e. _V -> A e. B ) <-> { A } C_ B ) -> ( A e. _V -> ( A e. B <-> { A } C_ B ) ) )
5	2 3 4	mpsyl	|- ( A e. V -> ( A e. B <-> { A } C_ B ) )