Theorem sucexb

Description: A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998)

	Ref	Expression
Assertion	sucexb	$$⊢A\in \mathrm{V}\leftrightarrow \mathrm{suc}A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		unexb	$$⊢\left(A\in \mathrm{V}\wedge \left\{A\right\}\in \mathrm{V}\right)\leftrightarrow A\cup \left\{A\right\}\in \mathrm{V}$$
2		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
3	2	biantru	$$⊢A\in \mathrm{V}\leftrightarrow \left(A\in \mathrm{V}\wedge \left\{A\right\}\in \mathrm{V}\right)$$
4		df-suc	$$⊢\mathrm{suc}A=A\cup \left\{A\right\}$$
5	4	eleq1i	$$⊢\mathrm{suc}A\in \mathrm{V}\leftrightarrow A\cup \left\{A\right\}\in \mathrm{V}$$
6	1 3 5	3bitr4i	$$⊢A\in \mathrm{V}\leftrightarrow \mathrm{suc}A\in \mathrm{V}$$