Mathlib Map

Theorems · Theorem · order theory

Set.singleton_ne_empty

∀ {α : Type u_1} (a : α), {a} ≠ ∅
Defined in
Mathlib.Data.Set.Insert
Cited by
8 results in Mathlib
Foundations
Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Set.isAtom_iff · cited by 2Set.isAtom_iffCovBy.exists_set_sdiff_singleton · cited by 2CovBy.exists_set_sdiff_si…Set.isAtom_singleton · cited by 1Set.isAtom_singletonIsMaxAntichain.isEmpty_iff · cited by 1IsMaxAntichain.isEmpty_iffSet.empty_ne_singleton · cited by 1Set.empty_ne_singletonSet.sdiff_singleton_covBy · cited by 1Set.sdiff_singleton_covByIsMaxChain.isEmpty_iff · cited by 1IsMaxChain.isEmpty_iffSet.support_subset_subsingleton_of_disjoint_on_support · cited by 1Set.support_subset_subsin…Set · cited by 53352SetSet.Nonempty.ne_empty · cited by 65Nonempty.ne_emptySet.singleton_nonempty · cited by 32Set.singleton_nonemptySet.singleton_ne_emptyCITED BYCITES

Cites3

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by8

Results whose statement or proof uses this declaration.