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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 7164 | . . 3 | |
2 | elioc1 8945 | . . 3 | |
3 | 1, 2 | sylan2 280 | . 2 |
4 | mnfxr 8848 | . . . . . . . 8 | |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | simpll 495 | . . . . . . 7 | |
7 | simpr1 944 | . . . . . . 7 | |
8 | mnfle 8867 | . . . . . . . 8 | |
9 | 8 | ad2antrr 471 | . . . . . . 7 |
10 | simpr2 945 | . . . . . . 7 | |
11 | 5, 6, 7, 9, 10 | xrlelttrd 8880 | . . . . . 6 |
12 | 1 | ad2antlr 472 | . . . . . . 7 |
13 | pnfxr 8846 | . . . . . . . 8 | |
14 | 13 | a1i 9 | . . . . . . 7 |
15 | simpr3 946 | . . . . . . 7 | |
16 | ltpnf 8856 | . . . . . . . 8 | |
17 | 16 | ad2antlr 472 | . . . . . . 7 |
18 | 7, 12, 14, 15, 17 | xrlelttrd 8880 | . . . . . 6 |
19 | xrrebnd 8886 | . . . . . . 7 | |
20 | 7, 19 | syl 14 | . . . . . 6 |
21 | 11, 18, 20 | mpbir2and 885 | . . . . 5 |
22 | 21, 10, 15 | 3jca 1118 | . . . 4 |
23 | 22 | ex 113 | . . 3 |
24 | rexr 7164 | . . . 4 | |
25 | 24 | 3anim1i 1124 | . . 3 |
26 | 23, 25 | impbid1 140 | . 2 |
27 | 3, 26 | bitrd 186 | 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 cpnf 7150 cmnf 7151 cxr 7152 clt 7153 cle 7154 cioc 8912 |
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-ioc 8916 |
This theorem is referenced by: iocssre 8976 |
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