range_ofENat — Mathlib

English

The image of ENat in Cardinal equals the initial segment up to aleph0: range ofENat = [0, aleph0].

Русский

Образ ENat в кардиналах равен начему отрезку до aleph0: range ofENat = [0, aleph0].

LaTeX

$$range\ ofENat = Iic\aleph0$$

Lean4

@[simp]
theorem range_ofENat : range ofENat = Iic ℵ₀ :=
  by
  refine (range_subset_iff.2 ofENat_le_aleph0).antisymm fun x (hx : x ≤ ℵ₀) ↦ ?_
  rcases hx.lt_or_eq with hlt | rfl
  · lift x to ℕ using hlt
    exact mem_range_self (x : ℕ∞)
  · exact mem_range_self (⊤ : ℕ∞)

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