Theorems · Theorem · order theory
lt_iff_le_not_ge
∀ {α : Type u_1} [inst : Preorder α] {a b : α}, a < b ↔ a ≤ b ∧ ¬b ≤ a- Defined in
- Mathlib.Order.Defs.PartialOrder
- Cited by
- 35 results in Mathlib
- Foundations
- Depth 3 from the axioms · uses no axioms
- Assumes
- Preorder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites2
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Preorderstatement and proof · cited by 7,952
- Preorder.lt_iff_le_not_geproof · cited by 1
Cited by35
Results whose statement or proof uses this declaration.
- le_of_ltproof · cited by 1,175
- not_le_of_gtproof · cited by 97
- lt_of_le_not_geproof · cited by 33
- lt_iff_lt_of_le_iff_le'proof · cited by 26
- not_isMax_iffproof · cited by 24
- SetLike.lt_iff_le_and_existsproof · cited by 14
- StrictMonoOn.lt_iff_ltproof · cited by 14
- maximal_iff_forall_gtproof · cited by 4
- Ideal.span_singleton_lt_span_singletonproof · cited by 4
- Concept.intent_ssubset_intent_iffproof · cited by 3
- Submodule.lt_map_of_comap_lt_of_surjectiveproof · cited by 2
- not_maximal_iff_exists_gtproof · cited by 2