Theorem onsupintrab2

Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025)

	Ref	Expression
Assertion	onsupintrab2	⊢ ( 𝐴 ∈ 𝒫 On → sup ( 𝐴 , On , E ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )

Proof

Step	Hyp	Ref	Expression
1		elpwb	⊢ ( 𝐴 ∈ 𝒫 On ↔ ( 𝐴 ∈ V ∧ 𝐴 ⊆ On ) )
2		onsupintrab	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ V ) → sup ( 𝐴 , On , E ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )
3	2	ancoms	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ⊆ On ) → sup ( 𝐴 , On , E ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )
4	1 3	sylbi	⊢ ( 𝐴 ∈ 𝒫 On → sup ( 𝐴 , On , E ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )