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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 523 | . . . . . 6 | |
2 | iooss1 9699 | . . . . . 6 | |
3 | 1, 2 | sylancom 416 | . . . . 5 |
4 | ioossicc 9742 | . . . . 5 | |
5 | 3, 4 | sstrdi 3109 | . . . 4 |
6 | sslin 3302 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | simplll 522 | . . . 4 | |
9 | simplrr 525 | . . . 4 | |
10 | df-ioo 9675 | . . . . 5 | |
11 | df-icc 9678 | . . . . 5 | |
12 | xrlenlt 7829 | . . . . 5 | |
13 | 10, 11, 12 | ixxdisj 9686 | . . . 4 |
14 | 8, 1, 9, 13 | syl3anc 1216 | . . 3 |
15 | 7, 14 | sseqtrd 3135 | . 2 |
16 | ss0 3403 | . 2 | |
17 | 15, 16 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cin 3070 wss 3071 c0 3363 class class class wbr 3929 (class class class)co 5774 cxr 7799 clt 7800 cle 7801 cioo 9671 cicc 9674 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-ioo 9675 df-icc 9678 |
This theorem is referenced by: (None) |
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