have hzmem : z ∈ A := by
  simp only [ne_eq, Subtype.exists, Set.mem_setOf_eq, not_true_eq_false, IsEmpty.forall_iff]
n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
⊢ ∃ m, n = S m
n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
⊢ ∃ m, n = S m