Theorems · Theorem · order theory
Set.isSimpleOrder_Iic_iff_isAtom
∀ {α : Type u_2} [inst : PartialOrder α] [inst_1 : OrderBot α] {a : α}, IsSimpleOrder ↑(Set.Iic a) ↔ IsAtom a- Defined in
- Mathlib.Order.Atoms
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PartialOrderOrderBot
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Top.topproof · cited by 9,680
- Set.Elemstatement and proof · cited by 7,166
- PartialOrderstatement and proof · cited by 6,410
- Bot.botproof · cited by 4,720
- le_of_ltproof · cited by 1,175
- Set.Iicstatement and proof · cited by 1,111
- OrderBotstatement and proof · cited by 1,055
- IsAtomstatement · cited by 130
- IsSimpleOrderstatement · cited by 54
- Subtype.mk_eq_mkproof · cited by 33
- Subtype.mk_lt_mkproof · cited by 8
Cited by3
Results whose statement or proof uses this declaration.
- isSimpleModule_iff_isAtomproof · cited by 7
- IsCompl.isAtom_iff_isCoatomproof · cited by 3
- CategoryTheory.subobject_simple_iff_isAtomproof · cited by 1