Theorem selsALT

Description: Alternate proof of sels , requiring ax-sep but not using el (which is proved from it as elALT ). (especially when the proof of el is inlined in sels ). (Contributed by NM, 4-Jan-2002) Generalize from the proof of elALT . (Revised by BJ, 3-Apr-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	selsALT	|- ( A e. V -> E. x A e. x )

Proof

Step	Hyp	Ref	Expression
1		snidg	|- ( A e. V -> A e. { A } )
2		snexg	|- ( A e. { A } -> { A } e. _V )
3		snidg	|- ( A e. { A } -> A e. { A } )
4		eleq2	|- ( x = { A } -> ( A e. x <-> A e. { A } ) )
5	2 3 4	spcedv	|- ( A e. { A } -> E. x A e. x )
6	1 5	syl	|- ( A e. V -> E. x A e. x )