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Mirrors > Home > ILE Home > Th. List > iooval2 | Unicode version |
Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
iooval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval 8931 | . 2 | |
2 | inrab2 3237 | . . . 4 | |
3 | ressxr 7162 | . . . . . 6 | |
4 | sseqin2 3185 | . . . . . 6 | |
5 | 3, 4 | mpbi 143 | . . . . 5 |
6 | rabeq 2595 | . . . . 5 | |
7 | 5, 6 | ax-mp 7 | . . . 4 |
8 | 2, 7 | eqtri 2101 | . . 3 |
9 | elioore 8935 | . . . . . 6 | |
10 | 9 | ssriv 3003 | . . . . 5 |
11 | 1, 10 | syl6eqssr 3050 | . . . 4 |
12 | df-ss 2986 | . . . 4 | |
13 | 11, 12 | sylib 120 | . . 3 |
14 | 8, 13 | syl5reqr 2128 | . 2 |
15 | 1, 14 | eqtrd 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 crab 2352 cin 2972 wss 2973 class class class wbr 3785 (class class class)co 5532 cr 6980 cxr 7152 clt 7153 cioo 8911 |
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-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
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-ioo 8915 |
This theorem is referenced by: elioo2 8944 ioomax 8971 ioopos 8973 dfioo2 8997 |
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