Theorem snelpwg

Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998) Put in closed form and avoid ax-nul . (Revised by BJ, 17-Jan-2025)

	Ref	Expression
Assertion	snelpwg	|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )

Proof

Step	Hyp	Ref	Expression
1		snssg	|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
2		snexg	|- ( A e. V -> { A } e. _V )
3		elpwg	|- ( { A } e. _V -> ( { A } e. ~P B <-> { A } C_ B ) )
4	2 3	syl	|- ( A e. V -> ( { A } e. ~P B <-> { A } C_ B ) )
5	1 4	bitr4d	|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )