Theorem elsuc2g

Description: Variant of membership in a successor, requiring that B rather than A be a set. (Contributed by NM, 28-Oct-2003)

	Ref	Expression
Assertion	elsuc2g	|- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-suc	|- suc B = ( B u. { B } )
2	1	eleq2i	|- ( A e. suc B <-> A e. ( B u. { B } ) )
3		elun	|- ( A e. ( B u. { B } ) <-> ( A e. B \/ A e. { B } ) )
4		elsn2g	|- ( B e. V -> ( A e. { B } <-> A = B ) )
5	4	orbi2d	|- ( B e. V -> ( ( A e. B \/ A e. { B } ) <-> ( A e. B \/ A = B ) ) )
6	3 5	syl5bb	|- ( B e. V -> ( A e. ( B u. { B } ) <-> ( A e. B \/ A = B ) ) )
7	2 6	syl5bb	|- ( B e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )