Theorem onuniintrab

Description: The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Closed form of uniordint . (Contributed by RP, 28-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ssonuni	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ⊆ On → ∪ 𝐴 ∈ On ) )
2	1	impcom	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∪ 𝐴 ∈ On )
3		intmin	⊢ ( ∪ 𝐴 ∈ On → ∩ { 𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥 } = ∪ 𝐴 )
4		unissb	⊢ ( ∪ 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 )
5	4	rabbii	⊢ { 𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥 } = { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 }
6	5	inteqi	⊢ ∩ { 𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥 } = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 }
7	3 6	eqtr3di	⊢ ( ∪ 𝐴 ∈ On → ∪ 𝐴 = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )
8	2 7	syl	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ∪ 𝐴 = ∩ { 𝑥 ∈ On ∣ ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 } )