Theorems · Theorem · combinatorics
Matroid.closure_empty_eq_ground_iff
∀ {α : Type u_2} {M : Matroid α}, M.closure ∅ = M.E ↔ M = Matroid.loopyOn M.E- Defined in
- Mathlib.Combinatorics.Matroid.Closure
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Matroidstatement and proof · cited by 1,258
- Matroid.Estatement and proof · cited by 550
- Matroid.closurestatement and proof · cited by 272
- subset_antisymmproof · cited by 150
- Set.empty_subsetproof · cited by 70
- Matroid.loopyOnstatement and proof · cited by 32
- Matroid.loopyOn_closure_eqproof · cited by 5
- Matroid.closure_monoproof · cited by 4
- Matroid.loopyOn_groundproof · cited by 3
- Matroid.ext_closureproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Matroid.eRank_eq_zero_iffproof · cited by 2
- Matroid.eq_loopyOn_iff_loops_eqproof · cited by 1
- Matroid.eq_loopyOn_iff_loopsproof · cited by 1