Theorem onsupintrab

Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of Schloeder p. 5. (Contributed by RP, 23-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		onsupuni	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → sup ( 𝐴 , On , E ) = ∪ 𝐴 )
2		onuniintrab	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∪ 𝐴 = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )
3	1 2	eqtrd	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → sup ( 𝐴 , On , E ) = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )