Theorems · Theorem · order theory
bot_ne_top
∀ {α : Type u} [inst : PartialOrder α] [inst_1 : BoundedOrder α] [Nontrivial α], ⊥ ≠ ⊤- Defined in
- Mathlib.Order.BoundedOrder.Basic
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topstatement and proof · cited by 9,680
- PartialOrderstatement and proof · cited by 6,410
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- BoundedOrderstatement and proof · cited by 270
- not_subsingletonproof · cited by 24
- subsingleton_of_bot_eq_topproof · cited by 8
Cited by24
Results whose statement or proof uses this declaration.
- IsSimpleModule.nontrivialproof · cited by 8
- bot_lt_topproof · cited by 7
- Ideal.exists_maximalproof · cited by 6
- LieModule.nontrivial_of_isIrreducibleproof · cited by 2
- IsLocalRing.exists_maximalIdeal_pow_le_of_isArtinianRing_quotientproof · cited by 2
- Module.jacobson_lt_topproof · cited by 2
- Ring.exists_maximal_of_not_isFieldproof · cited by 2
- Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRingproof · cited by 1
- Subring.exists_le_valuationSubring_of_isIntegrallyClosedInproof · cited by 1
- Profinite.exists_locallyConstant_finite_nonemptyproof · cited by 1
- Algebra.FormallyUnramified.isField_quotient_map_maximalIdealproof · cited by 1
- IsLocalRing.CotangentSpace.map_eq_top_iffproof · cited by 1