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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 13247 | . . . 4 ⊢ (𝐴 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) ∧ (𝐴 < 𝐴 ∧ 𝐴 < 𝐵))) | |
2 | 1 | simprbi 497 | . . 3 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐴 ∧ 𝐴 < 𝐵)) |
3 | 2 | simpld 495 | . 2 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐴) |
4 | 1 | simplbi 498 | . . . 4 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*)) |
5 | 4 | simp3d 1144 | . . 3 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → 𝐴 ∈ ℝ*) |
6 | xrltnr 12994 | . . 3 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐴(,)𝐵) → ¬ 𝐴 < 𝐴) |
8 | 3, 7 | pm2.65i 193 | 1 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7351 ℝ*cxr 11146 < clt 11147 (,)cioo 13218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-ioo 13222 |
This theorem is referenced by: lhop1lem 25329 lhop1 25330 lhop 25332 iooinlbub 43640 lptioo1 43774 volico 44125 fourierdlem61 44309 |
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