Theorem suceldifsn0

Description: A successor that is a member of a set is also a member of that set with the empty set removed. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	suceldifsn0.1	|- A e. B
	suceldifsn0.2	|- A = suc C
Assertion	suceldifsn0	|- A e. ( B \ { (/) } )

Proof

Step	Hyp	Ref	Expression
1		suceldifsn0.1	|- A e. B
2		suceldifsn0.2	|- A = suc C
3		nsuceq0	|- suc C =/= (/)
4	2 3	eqnetri	|- A =/= (/)
5		eldifsn	|- ( A e. ( B \ { (/) } ) <-> ( A e. B /\ A =/= (/) ) )
6	1 4 5	mpbir2an	|- A e. ( B \ { (/) } )