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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 7164 | . . . . . 6 | |
2 | elico2 8960 | . . . . . 6 | |
3 | 1, 2 | sylan2 280 | . . . . 5 |
4 | 3 | biimpd 142 | . . . 4 |
5 | 4 | 3adant3 958 | . . 3 |
6 | 3anass 923 | . . 3 | |
7 | 5, 6 | syl6ib 159 | . 2 |
8 | leadd1 7534 | . . . . . . . . . 10 | |
9 | 8 | 3com12 1142 | . . . . . . . . 9 |
10 | 9 | 3expib 1141 | . . . . . . . 8 |
11 | 10 | com12 30 | . . . . . . 7 |
12 | 11 | 3adant2 957 | . . . . . 6 |
13 | 12 | imp 122 | . . . . 5 |
14 | ltadd1 7533 | . . . . . . . . 9 | |
15 | 14 | 3expib 1141 | . . . . . . . 8 |
16 | 15 | com12 30 | . . . . . . 7 |
17 | 16 | 3adant1 956 | . . . . . 6 |
18 | 17 | imp 122 | . . . . 5 |
19 | 13, 18 | anbi12d 456 | . . . 4 |
20 | 19 | pm5.32da 439 | . . 3 |
21 | readdcl 7099 | . . . . . . . 8 | |
22 | 21 | expcom 114 | . . . . . . 7 |
23 | 22 | anim1d 329 | . . . . . 6 |
24 | 3anass 923 | . . . . . 6 | |
25 | 23, 24 | syl6ibr 160 | . . . . 5 |
26 | 25 | 3ad2ant3 961 | . . . 4 |
27 | readdcl 7099 | . . . . . 6 | |
28 | 27 | 3adant2 957 | . . . . 5 |
29 | readdcl 7099 | . . . . . 6 | |
30 | 29 | 3adant1 956 | . . . . 5 |
31 | rexr 7164 | . . . . . . 7 | |
32 | elico2 8960 | . . . . . . 7 | |
33 | 31, 32 | sylan2 280 | . . . . . 6 |
34 | 33 | biimprd 156 | . . . . 5 |
35 | 28, 30, 34 | syl2anc 403 | . . . 4 |
36 | 26, 35 | syld 44 | . . 3 |
37 | 20, 36 | sylbid 148 | . 2 |
38 | 7, 37 | syld 44 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 caddc 6984 cxr 7152 clt 7153 cle 7154 cico 8913 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-ico 8917 |
This theorem is referenced by: icoshftf1o 9013 |
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