powerlt_aleph0 — Mathlib

English

For any α, β' with α infinite, the number of equivalences between α and β' equals the number between β' and α.

Русский

Для бесконечного α и β' количество видимых взаимооднозначностей между α и β' равно между β' и α.

LaTeX

$$$ #(\\alpha \\simeq \\beta') = #(\\beta' \\simeq \\alpha) $$$

Lean4

theorem powerlt_aleph0 {c : Cardinal} (h : ℵ₀ ≤ c) : c ^< ℵ₀ = c :=
  by
  apply le_antisymm
  · rw [powerlt_le]
    intro c'
    rw [lt_aleph0]
    rintro ⟨n, rfl⟩
    apply power_nat_le h
  convert le_powerlt c one_lt_aleph0; rw [power_one]

Похожие утверждения