Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12755 | . . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐵))) | |
2 | 1 | simprbi 497 | . . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)) |
3 | 2 | simprd 496 | . 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵) |
4 | 1 | simplbi 498 | . . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
5 | 4 | simp2d 1135 | . . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ*) |
6 | xrltnr 12502 | . . 3 ⊢ (𝐵 ∈ ℝ* → ¬ 𝐵 < 𝐵) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → ¬ 𝐵 < 𝐵) |
8 | 3, 7 | pm2.65i 195 | 1 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝ*cxr 10662 < clt 10663 (,)cioo 12726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-ioo 12730 |
This theorem is referenced by: lhop 24540 iooinlbub 41652 lptioo2 41788 volioc 42133 fourierdlem60 42328 |
Copyright terms: Public domain | W3C validator |