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Mirrors > Home > MPE Home > Th. List > lbioo | Structured version Visualization version GIF version |
Description: An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Ref | Expression |
---|---|
lbioo | ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioo3g 12204 | . . . 4 ⊢ (𝐴 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) ∧ (𝐴 < 𝐴 ∧ 𝐴 < 𝐵))) | |
2 | 1 | simprbi 480 | . . 3 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐴 ∧ 𝐴 < 𝐵)) |
3 | 2 | simpld 475 | . 2 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐴) |
4 | 1 | simplbi 476 | . . . 4 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*)) |
5 | 4 | simp3d 1075 | . . 3 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → 𝐴 ∈ ℝ*) |
6 | xrltnr 11953 | . . 3 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → ¬ 𝐴 < 𝐴) |
8 | 3, 7 | pm2.65i 185 | 1 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝ*cxr 10073 < clt 10074 (,)cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-ioo 12179 |
This theorem is referenced by: lhop1lem 23776 lhop1 23777 lhop 23779 iooinlbub 39723 lptioo1 39864 volico 40200 fourierdlem61 40384 |
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