Theorem onsupneqmaxlim0

Description: If the supremum of a class of ordinals is not in that class, then the supremum is a limit ordinal or empty. (Contributed by RP, 27-Jan-2025)

	Ref	Expression
Assertion	onsupneqmaxlim0	⊢ ( 𝐴 ⊆ On → ( 𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		uniss	⊢ ( 𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴 )
2		ssorduni	⊢ ( 𝐴 ⊆ On → Ord ∪ 𝐴 )
3		orduniss	⊢ ( Ord ∪ 𝐴 → ∪ ∪ 𝐴 ⊆ ∪ 𝐴 )
4	2 3	syl	⊢ ( 𝐴 ⊆ On → ∪ ∪ 𝐴 ⊆ ∪ 𝐴 )
5	4	biantrud	⊢ ( 𝐴 ⊆ On → ( ∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ( ∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴 ) ) )
6		eqss	⊢ ( ∪ 𝐴 = ∪ ∪ 𝐴 ↔ ( ∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴 ) )
7	5 6	bitr4di	⊢ ( 𝐴 ⊆ On → ( ∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴 ) )
8	1 7	imbitrid	⊢ ( 𝐴 ⊆ On → ( 𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴 ) )