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	$$⊢A\in B$$
	suceldifsn0.2	$$⊢A=\mathrm{suc}C$$
Assertion	suceldifsn0	$$⊢A\in \left(B\setminus \left\{\varnothing \right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		suceldifsn0.1	$$⊢A\in B$$
2		suceldifsn0.2	$$⊢A=\mathrm{suc}C$$
3		nsuceq0	$$⊢\mathrm{suc}C\ne \varnothing$$
4	2 3	eqnetri	$$⊢A\ne \varnothing$$
5		eldifsn	$$⊢A\in \left(B\setminus \left\{\varnothing \right\}\right)\leftrightarrow \left(A\in B\wedge A\ne \varnothing \right)$$
6	1 4 5	mpbir2an	$$⊢A\in \left(B\setminus \left\{\varnothing \right\}\right)$$