/- Copyright 2025 The Formal Conjectures Authors. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at https://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -/ module public import Mathlib.Order.Interval.Finset.Basic public import Mathlib.Data.Set.Card@[expose] public sectiontheorem Finset.Iio_eventually_nonempty (β : Type*) [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : ∃ (b : β), ∀ n ≥ b, (Finset.Iio n).Nonempty := let ⟨b, hb⟩ := exists_ne (⊥ : β) ⟨b, fun n hn => byβ:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:b ≠ ⊥n:βhn:n ≥ b⊢ (Iio n).Nonempty simp [ne_bot_of_le_ne_bot hb hn]All goals completed! 🐙⟩theorem Finset.Iio_eventually_card_ne_zero (β : Type*) [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : ∃ (b : β), ∀ n ≥ b, (Finset.Iio n).card ≠ 0 := let ⟨b, hb⟩ := Iio_eventually_nonempty β ⟨b, fun n hn => byβ:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Iio n).Nonemptyn:βhn:n ≥ b⊢ #(Iio n) ≠ 0 simp [nonempty_iff_ne_empty.1 <| hb n hn]All goals completed! 🐙⟩ theorem Set.Iio_eventually_ncard_ne_zero (β : Type*) [PartialOrder β] [LocallyFiniteOrder β] [OrderBot β] [Nontrivial β] : ∃ (b : β), ∀ n ≥ b, (Set.Iio n).ncard ≠ 0 := let ⟨b, hb⟩ := Finset.Iio_eventually_card_ne_zero β ⟨b, fun n hn => byβ:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Finset.Iio n).card ≠ 0n:βhn:n ≥ b⊢ (Iio n).ncard ≠ 0 rw [← Finset.coe_Iio,β:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Finset.Iio n).card ≠ 0n:βhn:n ≥ b⊢ (↑(Finset.Iio n)).ncard ≠ 0β:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Finset.Iio n).card ≠ 0n:βhn:n ≥ b⊢ (Finset.Iio n).card ≠ 0 Set.ncard_coe_finsetβ:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Finset.Iio n).card ≠ 0n:βhn:n ≥ b⊢ (Finset.Iio n).card ≠ 0β:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Finset.Iio n).card ≠ 0n:βhn:n ≥ b⊢ (Finset.Iio n).card ≠ 0]β:Type u_1inst✝³:PartialOrder βinst✝²:LocallyFiniteOrder βinst✝¹:OrderBot βinst✝:Nontrivial βb:βhb:∀ n ≥ b, (Finset.Iio n).card ≠ 0n:βhn:n ≥ b⊢ (Finset.Iio n).card ≠ 0; exact hb n hnAll goals completed! 🐙⟩