Theorems · Theorem · order theory
Set.Finite.bddBelow_biUnion
∀ {α : Type u} {β : Type v} [inst : Preorder α] [IsCodirectedOrder α] [Nonempty α] {I : Set β} {S : β → Set α},
I.Finite → (BddBelow (⋃ i ∈ I, S i) ↔ ∀ i ∈ I, BddBelow (S i))A finite union of sets which are all bounded below is still bounded below.
- Defined in
- Mathlib.Data.Set.Finite.Lattice
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Preorderstatement and proof · cited by 7,952
- Set.iUnionstatement and proof · cited by 2,483
- Set.Finitestatement and proof · cited by 1,814
- BddBelowstatement and proof · cited by 401
- IsCodirectedOrderstatement and proof · cited by 95
- Set.Finite.induction_onproof · cited by 39
- Set.biUnion_insertproof · cited by 13
- Set.biUnion_emptyproof · cited by 12
- bddBelow_emptyproof · cited by 5
- Set.forall_mem_insertproof · cited by 4
- Set.forall_mem_emptyproof · cited by 3
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