Theorem sucidg

Description: Part of Proposition 7.23 of TakeutiZaring p. 41 (generalized). Lemma 1.7 of Schloeder p. 1. (Contributed by NM, 25-Mar-1995) (Proof shortened by Scott Fenton, 20-Feb-2012)

	Ref	Expression
Assertion	sucidg	$$⊢A\in V\to A\in \mathrm{suc}A$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢A=A$$
2	1	olci	$$⊢\left(A\in A\vee A=A\right)$$
3		elsucg	$$⊢A\in V\to \left(A\in \mathrm{suc}A\leftrightarrow \left(A\in A\vee A=A\right)\right)$$
4	2 3	mpbiri	$$⊢A\in V\to A\in \mathrm{suc}A$$