theorem extracted_formal_statement_2.{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 (dist 0 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 (dist 0 x) ≥ EMetric.infEdist 0 s

Tactic: apply (ENNReal.le_ofReal_iff_toReal_le _ _).2

State after:
[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
 ⊢ (EMetric.infEdist 0 s).toReal ≤ dist 0 x,
 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:
ENNReal.le_ofReal_iff_toReal_le : ∀ {a : ENNReal} {b : ℝ}, a ≠ ⊤ → 0 ≤ b → (a ≤ ENNReal.ofReal b ↔ a.toReal ≤ b)