Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eliooord | Structured version Visualization version GIF version |
Description: Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.) |
Ref | Expression |
---|---|
eliooord | ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliooxr 12993 | . . . 4 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) | |
2 | elioo2 12976 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
4 | 3 | ibi 270 | . 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
5 | 3simpc 1152 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 ℝ*cxr 10866 < clt 10867 (,)cioo 12935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-ioo 12939 |
This theorem is referenced by: elioo4g 12995 iccssioo2 13008 qdensere 23667 zcld 23710 reconnlem2 23724 xrge0tsms 23731 ovolioo 24465 ioorcl2 24469 itgsplitioo 24735 dvferm1lem 24881 dvferm2lem 24883 dvferm 24885 dvlt0 24902 dvivthlem1 24905 lhop1lem 24910 lhop1 24911 lhop2 24912 dvcvx 24917 ftc1lem4 24936 itgsubstlem 24945 itgsubst 24946 pilem2 25344 pilem3 25345 pigt2lt4 25346 tangtx 25395 tanabsge 25396 cosne0 25418 cos0pilt1 25421 tanord 25427 tanregt0 25428 argimlt0 25501 logneg2 25503 divlogrlim 25523 logno1 25524 logcnlem3 25532 dvloglem 25536 logf1o2 25538 loglesqrt 25644 asinsin 25775 acoscos 25776 atanlogaddlem 25796 atanlogsub 25799 atantan 25806 atanbndlem 25808 scvxcvx 25868 lgamgulmlem2 25912 basellem8 25970 vmalogdivsum2 26419 vmalogdivsum 26420 2vmadivsumlem 26421 chpdifbndlem1 26434 selberg3lem1 26438 selberg3 26440 selberg4lem1 26441 selberg4 26442 selberg3r 26450 selberg4r 26451 selberg34r 26452 pntrlog2bndlem1 26458 pntrlog2bndlem2 26459 pntrlog2bndlem3 26460 pntrlog2bndlem4 26461 pntrlog2bndlem5 26462 pntrlog2bndlem6a 26463 pntrlog2bndlem6 26464 pntrlog2bnd 26465 pntpbnd1a 26466 pntpbnd1 26467 pntpbnd2 26468 pntpbnd 26469 pntibndlem2 26472 pntibndlem3 26473 pntibnd 26474 pntlemd 26475 pntlemb 26478 pntlemr 26483 pnt 26495 padicabv 26511 xrge0tsmsd 31036 fct2relem 32289 logdivsqrle 32342 knoppndvlem3 34431 iooelexlt 35270 relowlssretop 35271 poimir 35547 itg2gt0cn 35569 ftc1cnnclem 35585 aks4d1p1p5 39816 radcnvrat 41605 cncfiooicclem1 43109 itgioocnicc 43193 iblcncfioo 43194 amgmwlem 46177 |
Copyright terms: Public domain | W3C validator |