Theorems · Theorem · order theory
LE.le.lt_of_ne
∀ {α : Type u_1} [inst : PartialOrder α] {a b : α}, a ≤ b → a ≠ b → a < bAlias of lt_of_le_of_ne.
- Defined in
- Mathlib.Order.Basic
- Cited by
- 116 results in Mathlib
- Foundations
- Depth 6 from the axioms, rests on 15 definitions · uses no axioms
- Assumes
- PartialOrder
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.
- PartialOrderstatement · cited by 6,410
- lt_of_le_of_neproof · cited by 230
Cited by116
Results whose statement or proof uses this declaration.
- zero_lt_oneproof · cited by 598
- lt_iff_le_and_neproof · cited by 47
- LE.le.lt_iff_neproof · cited by 34
- Monotone.strictMono_of_injectiveproof · cited by 22
- closure_ballproof · cited by 20
- LE.le.ssubset_of_neproof · cited by 16
- ENNReal.ofReal_rpow_of_nonnegproof · cited by 13
- Sbtw.angle₁₂₃_eq_piproof · cited by 6
- Filter.Tendsto.rpow_constproof · cited by 6
- Wbtw.angle₂₁₃_eq_zero_of_neproof · cited by 5
- IsLUB.exists_seq_strictMono_tendsto_of_notMemproof · cited by 4
- Set.Finite.encard_lt_encardproof · cited by 4