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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 7071 | . . 3 | |
2 | elioc1 8791 | . . 3 | |
3 | 1, 2 | sylan2 270 | . 2 |
4 | mnfxr 8694 | . . . . . . . 8 | |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | simpll 481 | . . . . . . 7 | |
7 | simpr1 910 | . . . . . . 7 | |
8 | mnfle 8713 | . . . . . . . 8 | |
9 | 8 | ad2antrr 457 | . . . . . . 7 |
10 | simpr2 911 | . . . . . . 7 | |
11 | 5, 6, 7, 9, 10 | xrlelttrd 8726 | . . . . . 6 |
12 | 1 | ad2antlr 458 | . . . . . . 7 |
13 | pnfxr 8692 | . . . . . . . 8 | |
14 | 13 | a1i 9 | . . . . . . 7 |
15 | simpr3 912 | . . . . . . 7 | |
16 | ltpnf 8702 | . . . . . . . 8 | |
17 | 16 | ad2antlr 458 | . . . . . . 7 |
18 | 7, 12, 14, 15, 17 | xrlelttrd 8726 | . . . . . 6 |
19 | xrrebnd 8732 | . . . . . . 7 | |
20 | 7, 19 | syl 14 | . . . . . 6 |
21 | 11, 18, 20 | mpbir2and 851 | . . . . 5 |
22 | 21, 10, 15 | 3jca 1084 | . . . 4 |
23 | 22 | ex 108 | . . 3 |
24 | rexr 7071 | . . . 4 | |
25 | 24 | 3anim1i 1090 | . . 3 |
26 | 23, 25 | impbid1 130 | . 2 |
27 | 3, 26 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 cle 7061 cioc 8758 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-po 4033 df-iso 4034 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-ioc 8762 |
This theorem is referenced by: iocssre 8822 |
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