Theorems · Theorem · order theory
lt_of_le_of_ne
∀ {α : Type u_1} [inst : PartialOrder α] {a b : α}, a ≤ b → a ≠ b → a < b- Defined in
- Mathlib.Order.Defs.PartialOrder
- Cited by
- 230 results in Mathlib
- Foundations
- Depth 5 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.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- PartialOrderstatement and proof · cited by 6,410
- le_antisymmproof · cited by 2,068
- lt_of_le_not_geproof · cited by 33
Cited by230
Results whose statement or proof uses this declaration.
- LE.le.lt_of_neproof · cited by 116
- Ne.lt_of_leproof · cited by 27
- ENNReal.ofReal_rpow_of_nonnegproof · cited by 13
- Commute.add_powproof · cited by 12
- NormedAddGroupHom.le_opNormproof · cited by 9
- Ideal.comap_isMaximal_of_surjectiveproof · cited by 8
- Ne.le_iff_ltproof · cited by 8
- Polynomial.coeff_comp_degree_mul_degreeproof · cited by 7
- ExistsContDiffBumpBase.u_existsproof · cited by 5
- EReal.strictMono_div_right_of_posproof · cited by 5
- ZLattice.covolume_posproof · cited by 5
- Real.geom_mean_eq_arith_mean_weighted_iff_of_pos'proof · cited by 4
Showing the 200 most cited of 230.