Theorem onunisuci

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994)

	Ref	Expression
Hypothesis	on.1	⊢ 𝐴 ∈ On
Assertion	onunisuci	⊢ ∪ suc 𝐴 = 𝐴

Proof

Step	Hyp	Ref	Expression
1		on.1	⊢ 𝐴 ∈ On
2	1	ontrci	⊢ Tr 𝐴
3	1	elexi	⊢ 𝐴 ∈ V
4	3	unisuc	⊢ ( Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴 )
5	2 4	mpbi	⊢ ∪ suc 𝐴 = 𝐴