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Mirrors > Home > ILE Home > Th. List > ioodisj | Unicode version |
Description: If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.) |
Ref | Expression |
---|---|
ioodisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 524 | . . . . . 6 | |
2 | iooss1 9820 | . . . . . 6 | |
3 | 1, 2 | sylancom 417 | . . . . 5 |
4 | ioossicc 9863 | . . . . 5 | |
5 | 3, 4 | sstrdi 3140 | . . . 4 |
6 | sslin 3333 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | simplll 523 | . . . 4 | |
9 | simplrr 526 | . . . 4 | |
10 | df-ioo 9796 | . . . . 5 | |
11 | df-icc 9799 | . . . . 5 | |
12 | xrlenlt 7942 | . . . . 5 | |
13 | 10, 11, 12 | ixxdisj 9807 | . . . 4 |
14 | 8, 1, 9, 13 | syl3anc 1220 | . . 3 |
15 | 7, 14 | sseqtrd 3166 | . 2 |
16 | ss0 3434 | . 2 | |
17 | 15, 16 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cin 3101 wss 3102 c0 3394 class class class wbr 3965 (class class class)co 5824 cxr 7911 clt 7912 cle 7913 cioo 9792 cicc 9795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-po 4256 df-iso 4257 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-ioo 9796 df-icc 9799 |
This theorem is referenced by: (None) |
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