Theorem onsupex3

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

	Ref	Expression
Assertion	onsupex3	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		onsupcl3	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } ∈ On )
2	1	elexd	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } ∈ V )