Theorem onuniorsuc

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) Put in closed form. (Revised by BJ, 11-Jan-2025)

	Ref	Expression
Assertion	onuniorsuc	$$⊢A\in \mathrm{On}\to \left(A=\bigcup A\vee A=\mathrm{suc}\bigcup A\right)$$

Proof

Step	Hyp	Ref	Expression
1		eloni	$$⊢A\in \mathrm{On}\to \mathrm{Ord}A$$
2		orduniorsuc	$$⊢\mathrm{Ord}A\to \left(A=\bigcup A\vee A=\mathrm{suc}\bigcup A\right)$$
3	1 2	syl	$$⊢A\in \mathrm{On}\to \left(A=\bigcup A\vee A=\mathrm{suc}\bigcup A\right)$$