Theorem elsucg

Description: Membership in a successor. Exercise 5 of TakeutiZaring p. 17. (Contributed by NM, 15-Sep-1995)

	Ref	Expression
Assertion	elsucg	|- ( A 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	2 3	bitri	|- ( A e. suc B <-> ( A e. B \/ A e. { B } ) )
5		elsng	|- ( A e. V -> ( A e. { B } <-> A = B ) )
6	5	orbi2d	|- ( A e. V -> ( ( A e. B \/ A e. { B } ) <-> ( A e. B \/ A = B ) ) )
7	4 6	bitrid	|- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )