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Mirrors > Home > ILE Home > Th. List > icoshft | Unicode version |
Description: A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.) |
Ref | Expression |
---|---|
icoshft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7811 | . . . . . 6 | |
2 | elico2 9720 | . . . . . 6 | |
3 | 1, 2 | sylan2 284 | . . . . 5 |
4 | 3 | biimpd 143 | . . . 4 |
5 | 4 | 3adant3 1001 | . . 3 |
6 | 3anass 966 | . . 3 | |
7 | 5, 6 | syl6ib 160 | . 2 |
8 | leadd1 8192 | . . . . . . . . . 10 | |
9 | 8 | 3com12 1185 | . . . . . . . . 9 |
10 | 9 | 3expib 1184 | . . . . . . . 8 |
11 | 10 | com12 30 | . . . . . . 7 |
12 | 11 | 3adant2 1000 | . . . . . 6 |
13 | 12 | imp 123 | . . . . 5 |
14 | ltadd1 8191 | . . . . . . . . 9 | |
15 | 14 | 3expib 1184 | . . . . . . . 8 |
16 | 15 | com12 30 | . . . . . . 7 |
17 | 16 | 3adant1 999 | . . . . . 6 |
18 | 17 | imp 123 | . . . . 5 |
19 | 13, 18 | anbi12d 464 | . . . 4 |
20 | 19 | pm5.32da 447 | . . 3 |
21 | readdcl 7746 | . . . . . . . 8 | |
22 | 21 | expcom 115 | . . . . . . 7 |
23 | 22 | anim1d 334 | . . . . . 6 |
24 | 3anass 966 | . . . . . 6 | |
25 | 23, 24 | syl6ibr 161 | . . . . 5 |
26 | 25 | 3ad2ant3 1004 | . . . 4 |
27 | readdcl 7746 | . . . . . 6 | |
28 | 27 | 3adant2 1000 | . . . . 5 |
29 | readdcl 7746 | . . . . . 6 | |
30 | 29 | 3adant1 999 | . . . . 5 |
31 | rexr 7811 | . . . . . . 7 | |
32 | elico2 9720 | . . . . . . 7 | |
33 | 31, 32 | sylan2 284 | . . . . . 6 |
34 | 33 | biimprd 157 | . . . . 5 |
35 | 28, 30, 34 | syl2anc 408 | . . . 4 |
36 | 26, 35 | syld 45 | . . 3 |
37 | 20, 36 | sylbid 149 | . 2 |
38 | 7, 37 | syld 45 | 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 caddc 7623 cxr 7799 clt 7800 cle 7801 cico 9673 |
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-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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-ico 9677 |
This theorem is referenced by: icoshftf1o 9774 |
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