Metamath Proof Explorer


Theorem uniordint

Description: The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of BellMachover p. 471. (Contributed by NM, 20-Sep-2003)

Ref Expression
Hypothesis uniordint.1 𝐴 ∈ V
Assertion uniordint ( 𝐴 ⊆ On → 𝐴 = { 𝑥 ∈ On ∣ ∀ 𝑦𝐴 𝑦𝑥 } )

Proof

Step Hyp Ref Expression
1 uniordint.1 𝐴 ∈ V
2 1 ssonunii ( 𝐴 ⊆ On → 𝐴 ∈ On )
3 intmin ( 𝐴 ∈ On → { 𝑥 ∈ On ∣ 𝐴𝑥 } = 𝐴 )
4 unissb ( 𝐴𝑥 ↔ ∀ 𝑦𝐴 𝑦𝑥 )
5 4 rabbii { 𝑥 ∈ On ∣ 𝐴𝑥 } = { 𝑥 ∈ On ∣ ∀ 𝑦𝐴 𝑦𝑥 }
6 5 inteqi { 𝑥 ∈ On ∣ 𝐴𝑥 } = { 𝑥 ∈ On ∣ ∀ 𝑦𝐴 𝑦𝑥 }
7 3 6 eqtr3di ( 𝐴 ∈ On → 𝐴 = { 𝑥 ∈ On ∣ ∀ 𝑦𝐴 𝑦𝑥 } )
8 2 7 syl ( 𝐴 ⊆ On → 𝐴 = { 𝑥 ∈ On ∣ ∀ 𝑦𝐴 𝑦𝑥 } )