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 = 𝐴𝐴 = suc 𝐴 ) )

Proof

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