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 12761 | . . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐵))) | |
2 | 1 | simprbi 499 | . . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)) |
3 | 2 | simprd 498 | . 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵) |
4 | 1 | simplbi 500 | . . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
5 | 4 | simp2d 1139 | . . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ*) |
6 | xrltnr 12508 | . . 3 ⊢ (𝐵 ∈ ℝ* → ¬ 𝐵 < 𝐵) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → ¬ 𝐵 < 𝐵) |
8 | 3, 7 | pm2.65i 196 | 1 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝ*cxr 10668 < clt 10669 (,)cioo 12732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-ioo 12736 |
This theorem is referenced by: lhop 24607 iooinlbub 41769 lptioo2 41905 volioc 42250 fourierdlem60 42445 |
Copyright terms: Public domain | W3C validator |