theorem extracted_formal_statement_0.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] {s : Set E} (hs : s.Nonempty)
  (x : E) (xs : x ∈ s) (this : EMetric.infEdist 0 s ≤ edist 0 x)
  (h_original_goal : ENNReal.ofReal ‖x‖ ≥ EMetric.infEdist 0 s) : EMetric.infEdist 0 s ≠ ⊤ := sorry

State before:
case neg
E : Type u_1
inst✝ : NormedAddCommGroup E
s : Set E
hs : s.Nonempty
x : E
xs : x ∈ s
this : EMetric.infEdist 0 s ≤ edist 0 x
⊢ ENNReal.ofReal ‖x‖ ≥ EMetric.infEdist 0 s

Tactic: rw [← dist_zero_left]
    apply (ENNReal.le_ofReal_iff_toReal_le _ _).2
    · exact ENNReal.toReal_le_of_le_ofReal dist_nonneg (edist_dist 0 x ▸ this)

State after:
[E : Type u_1
 inst✝ : NormedAddCommGroup E
 s : Set E
 hs : s.Nonempty
 x : E
 xs : x ∈ s
 this : EMetric.infEdist 0 s ≤ edist 0 x
 ⊢ EMetric.infEdist 0 s ≠ ⊤,
 E : Type u_1
 inst✝ : NormedAddCommGroup E
 s : Set E
 hs : s.Nonempty
 x : E
 xs : x ∈ s
 this : EMetric.infEdist 0 s ≤ edist 0 x
 ⊢ 0 ≤ dist 0 x]

Relevant lemmas:
dist_nonneg : ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α}, 0 ≤ dist x y
dist_zero_left : ∀ {E : Type u_5} [inst : SeminormedAddGroup E] (a : E), dist 0 a = ‖a‖
ENNReal.toReal_le_of_le_ofReal : ∀ {a : ENNReal} {b : ℝ}, 0 ≤ b → a ≤ ENNReal.ofReal b → a.toReal ≤ b
ENNReal.le_ofReal_iff_toReal_le : ∀ {a : ENNReal} {b : ℝ}, a ≠ ⊤ → 0 ≤ b → (a ≤ ENNReal.ofReal b ↔ a.toReal ≤ b)
edist_dist : ∀ {α : Type u} [inst : PseudoMetricSpace α] (x y : α), edist x y = ENNReal.ofReal (dist x y)