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Metamath Proof Explorer

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 ∈ On → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2		orduniorsuc	⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ On → ( 𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴 ) )