-- I actually thought that ∀p, ∃q, (q ≠ p) ∧ q!
-- However, propositional extensionality tells us that (p ↔ q) → p = q

theorem finite_true_props
    : ¬∀p, ∃q, (q ≠ p) ∧ q := by
  intro h
  have := h True
  simp at this

theorem finite_false_props
    : ¬∀p, ∃q, (q ≠ p) ∧ ¬q := by
  intro h
  have := h False
  simp at this