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Mirrors > Home > MPE Home > Th. List > ubioo | Structured version Visualization version GIF version |
Description: An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Ref | Expression |
---|---|
ubioo | ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioo3g 12453 | . . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐵))) | |
2 | 1 | simprbi 491 | . . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)) |
3 | 2 | simprd 490 | . 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵) |
4 | 1 | simplbi 492 | . . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
5 | 4 | simp2d 1174 | . . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ*) |
6 | xrltnr 12200 | . . 3 ⊢ (𝐵 ∈ ℝ* → ¬ 𝐵 < 𝐵) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → ¬ 𝐵 < 𝐵) |
8 | 3, 7 | pm2.65i 186 | 1 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 385 ∧ w3a 1108 ∈ wcel 2157 class class class wbr 4843 (class class class)co 6878 ℝ*cxr 10362 < clt 10363 (,)cioo 12424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-pre-lttri 10298 ax-pre-lttrn 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-ioo 12428 |
This theorem is referenced by: lhop 24120 iooinlbub 40471 lptioo2 40607 volioc 40931 fourierdlem60 41126 |
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