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Mirrors > Home > MPE Home > Th. List > bndndx | Structured version Visualization version GIF version |
Description: A bounded real sequence 𝐴(𝑘) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.) |
Ref | Expression |
---|---|
bndndx | ⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arch 11166 | . . . 4 ⊢ (𝑥 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑥 < 𝑘) | |
2 | nnre 10904 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ) | |
3 | lelttr 10007 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝑘) → 𝐴 < 𝑘)) | |
4 | ltle 10005 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐴 < 𝑘 → 𝐴 ≤ 𝑘)) | |
5 | 4 | 3adant2 1073 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝐴 < 𝑘 → 𝐴 ≤ 𝑘)) |
6 | 3, 5 | syld 46 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝑘) → 𝐴 ≤ 𝑘)) |
7 | 6 | exp5o 1278 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝑥 ∈ ℝ → (𝑘 ∈ ℝ → (𝐴 ≤ 𝑥 → (𝑥 < 𝑘 → 𝐴 ≤ 𝑘))))) |
8 | 7 | com3l 87 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ ℝ → (𝐴 ∈ ℝ → (𝐴 ≤ 𝑥 → (𝑥 < 𝑘 → 𝐴 ≤ 𝑘))))) |
9 | 8 | imp4b 611 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → (𝑥 < 𝑘 → 𝐴 ≤ 𝑘))) |
10 | 9 | com23 84 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑥 < 𝑘 → ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘))) |
11 | 2, 10 | sylan2 490 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑥 < 𝑘 → ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘))) |
12 | 11 | reximdva 3000 | . . . 4 ⊢ (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ 𝑥 < 𝑘 → ∃𝑘 ∈ ℕ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘))) |
13 | 1, 12 | mpd 15 | . . 3 ⊢ (𝑥 ∈ ℝ → ∃𝑘 ∈ ℕ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘)) |
14 | r19.35 3065 | . . 3 ⊢ (∃𝑘 ∈ ℕ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → 𝐴 ≤ 𝑘) ↔ (∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘)) | |
15 | 13, 14 | sylib 207 | . 2 ⊢ (𝑥 ∈ ℝ → (∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘)) |
16 | 15 | rexlimiv 3009 | 1 ⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝑥) → ∃𝑘 ∈ ℕ 𝐴 ≤ 𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 ℝcr 9814 < clt 9953 ≤ cle 9954 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 |
This theorem is referenced by: (None) |
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