Theorem elsuc

Description: Membership in a successor. Exercise 5 of TakeutiZaring p. 17. (Contributed by NM, 15-Sep-2003)

	Ref	Expression
Hypothesis	elsuc.1	$$⊢A\in \mathrm{V}$$
Assertion	elsuc	$$⊢A\in \mathrm{suc}B\leftrightarrow \left(A\in B\vee A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		elsuc.1	$$⊢A\in \mathrm{V}$$
2		elsucg	$$⊢A\in \mathrm{V}\to \left(A\in \mathrm{suc}B\leftrightarrow \left(A\in B\vee A=B\right)\right)$$
3	1 2	ax-mp	$$⊢A\in \mathrm{suc}B\leftrightarrow \left(A\in B\vee A=B\right)$$