Theorem onsupcl2

Description: The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	onsupcl2	⊢ ( 𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On )

Proof

Step	Hyp	Ref	Expression
1		elpwb	⊢ ( 𝐴 ∈ 𝒫 On ↔ ( 𝐴 ∈ V ∧ 𝐴 ⊆ On ) )
2		ssonuni	⊢ ( 𝐴 ∈ V → ( 𝐴 ⊆ On → ∪ 𝐴 ∈ On ) )
3	2	imp	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ⊆ On ) → ∪ 𝐴 ∈ On )
4	1 3	sylbi	⊢ ( 𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On )