Theorem onsupcl3

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

	Ref	Expression
Assertion	onsupcl3	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } ∈ On )

Proof

Step	Hyp	Ref	Expression
1		onuniintrab	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∪ 𝐴 = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )
2		ssonuni	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ⊆ On → ∪ 𝐴 ∈ On ) )
3	2	impcom	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∪ 𝐴 ∈ On )
4	1 3	eqeltrrd	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } ∈ On )