Theorem ordunisuc

Description: An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	ordunisuc	⊢ ( Ord 𝐴 → ∪ suc 𝐴 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ordeleqon	⊢ ( Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On ) )
2		onunisuc	⊢ ( 𝐴 ∈ On → ∪ suc 𝐴 = 𝐴 )
3		sucon	⊢ suc On = On
4	3	unieqi	⊢ ∪ suc On = ∪ On
5		unon	⊢ ∪ On = On
6	4 5	eqtri	⊢ ∪ suc On = On
7		suceq	⊢ ( 𝐴 = On → suc 𝐴 = suc On )
8	7	unieqd	⊢ ( 𝐴 = On → ∪ suc 𝐴 = ∪ suc On )
9		id	⊢ ( 𝐴 = On → 𝐴 = On )
10	6 8 9	3eqtr4a	⊢ ( 𝐴 = On → ∪ suc 𝐴 = 𝐴 )
11	2 10	jaoi	⊢ ( ( 𝐴 ∈ On ∨ 𝐴 = On ) → ∪ suc 𝐴 = 𝐴 )
12	1 11	sylbi	⊢ ( Ord 𝐴 → ∪ suc 𝐴 = 𝐴 )