Theorem: Matroid.not_nonloop_iff_cl (Matroid/Loop.lean)

State before:
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
⊢ ¬M.Nonloop e ↔ M.cl {e} = M.cl ∅
Tactic:
by_cases he : e ∈ M.E
State after:
case pos
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : e ∈ M.E
⊢ ¬M.Nonloop e ↔ M.cl {e} = M.cl ∅

case neg
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : e ∉ M.E
⊢ ¬M.Nonloop e ↔ M.cl {e} = M.cl ∅

State before:
case neg
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : e ∉ M.E
⊢ ¬M.Nonloop e ↔ M.cl {e} = M.cl ∅
Tactic:
simp [← cl_inter_ground, singleton_inter_eq_empty.2 he,
  (show ¬ M.Nonloop e from fun h ↦ he h.mem_ground)]
State after:
no goals

State before:
case pos
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X Y Z F C K : Set α
he : e ∈ M.E
⊢ ¬M.Nonloop e ↔ M.cl {e} = M.cl ∅
Tactic:
simp [Nonloop, and_comm, not_and, not_not, loop_iff_cl_eq_cl_empty', he]
State after:
no goals