Theorem onuniorsuci

Description: An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994)

	Ref	Expression
Hypothesis	onssi.1	$$⊢A\in \mathrm{On}$$
Assertion	onuniorsuci	$$⊢\left(A=\bigcup A\vee A=\mathrm{suc}\bigcup A\right)$$

Proof

Step	Hyp	Ref	Expression
1		onssi.1	$$⊢A\in \mathrm{On}$$
2	1	onordi	$$⊢\mathrm{Ord}A$$
3		orduniorsuc	$$⊢\mathrm{Ord}A\to \left(A=\bigcup A\vee A=\mathrm{suc}\bigcup A\right)$$
4	2 3	ax-mp	$$⊢\left(A=\bigcup A\vee A=\mathrm{suc}\bigcup A\right)$$