Theorems · Theorem · order theory
le_iff_le_iff_lt_iff_lt
∀ {α : Type u_2} {β : Type u_5} [inst : LinearOrder α] [inst_1 : LinearOrder β] {a b : α} {c d : β},
(a ≤ b ↔ c ≤ d) ↔ (b < a ↔ d < c)- Defined in
- Mathlib.Order.Basic
- Cited by
- 25 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses no axioms
- Assumes
- LinearOrderLinearOrder
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.
- LinearOrderstatement and proof · cited by 8,572
- not_ltproof · cited by 306
- lt_iff_lt_of_le_iff_leproof · cited by 54
Cited by25
Results whose statement or proof uses this declaration.
- Real.rpow_le_rpow_iffproof · cited by 14
- Nat.le_log_iff_pow_leproof · cited by 5
- Real.le_sqrtproof · cited by 5
- Valuation.IsEquiv.lt_iff_ltproof · cited by 4
- AbsoluteValue.isEquiv_iff_lt_one_iffproof · cited by 3
- Ordinal.card_le_natproof · cited by 3
- Real.le_sqrt'proof · cited by 3
- Real.rpow_le_rpow_iff_of_negproof · cited by 3
- Order.IsSuccPrelimit.le_succ_iffproof · cited by 2
- Odd.pow_nonneg_iffproof · cited by 1
- Order.IsPredPrelimit.pred_le_iffproof · cited by 1
- ArchimedeanClass.mk_sub_lt_mk_left_iffproof · cited by 0