Theorem elsuci

Description: Membership in a successor. This one-way implication does not require that either A or B be sets. Lemma 1.13 of Schloeder p. 2. (Contributed by NM, 6-Jun-1994)

	Ref	Expression
Assertion	elsuci	|- ( 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		elsni	|- ( A e. { B } -> A = B )
6	5	orim2i	|- ( ( A e. B \/ A e. { B } ) -> ( A e. B \/ A = B ) )
7	4 6	sylbi	|- ( A e. suc B -> ( A e. B \/ A = B ) )