Theorem snex

Description: A singleton is a set. Theorem 7.12 of Quine p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT . (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 19-May-2013)

	Ref	Expression
Assertion	snex	|- { A } e. _V

Proof

Step	Hyp	Ref	Expression
1		snexg	|- ( A e. _V -> { A } e. _V )
2		snprc	|- ( -. A e. _V <-> { A } = (/) )
3	2	biimpi	|- ( -. A e. _V -> { A } = (/) )
4		0ex	|- (/) e. _V
5	3 4	eqeltrdi	|- ( -. A e. _V -> { A } e. _V )
6	1 5	pm2.61i	|- { A } e. _V