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Mirrors > Home > ILE Home > Th. List > elioc2 | Unicode version |
Description: Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
elioc2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7811 | . . 3 | |
2 | elioc1 9705 | . . 3 | |
3 | 1, 2 | sylan2 284 | . 2 |
4 | mnfxr 7822 | . . . . . . . 8 | |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | simpll 518 | . . . . . . 7 | |
7 | simpr1 987 | . . . . . . 7 | |
8 | mnfle 9578 | . . . . . . . 8 | |
9 | 8 | ad2antrr 479 | . . . . . . 7 |
10 | simpr2 988 | . . . . . . 7 | |
11 | 5, 6, 7, 9, 10 | xrlelttrd 9593 | . . . . . 6 |
12 | 1 | ad2antlr 480 | . . . . . . 7 |
13 | pnfxr 7818 | . . . . . . . 8 | |
14 | 13 | a1i 9 | . . . . . . 7 |
15 | simpr3 989 | . . . . . . 7 | |
16 | ltpnf 9567 | . . . . . . . 8 | |
17 | 16 | ad2antlr 480 | . . . . . . 7 |
18 | 7, 12, 14, 15, 17 | xrlelttrd 9593 | . . . . . 6 |
19 | xrrebnd 9602 | . . . . . . 7 | |
20 | 7, 19 | syl 14 | . . . . . 6 |
21 | 11, 18, 20 | mpbir2and 928 | . . . . 5 |
22 | 21, 10, 15 | 3jca 1161 | . . . 4 |
23 | 22 | ex 114 | . . 3 |
24 | rexr 7811 | . . . 4 | |
25 | 24 | 3anim1i 1167 | . . 3 |
26 | 23, 25 | impbid1 141 | . 2 |
27 | 3, 26 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 cle 7801 cioc 9672 |
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-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-ioc 9676 |
This theorem is referenced by: iocssre 9736 ef01bndlem 11463 sin01bnd 11464 cos01bnd 11465 cos1bnd 11466 sin01gt0 11468 cos01gt0 11469 sin02gt0 11470 sincos1sgn 11471 sincos2sgn 11472 cos12dec 11474 sin0pilem1 12862 sin0pilem2 12863 sinhalfpilem 12872 sincosq1lem 12906 coseq0negpitopi 12917 tangtx 12919 sincos4thpi 12921 pigt3 12925 |
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