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 ∈ On
Assertion	onuniorsuci	⊢ ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		onssi.1	⊢ 𝐴 ∈ On
2	1	onordi	⊢ Ord 𝐴
3		orduniorsuc	⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 ) )
4	2 3	ax-mp	⊢ ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 )