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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7997 |
. . 3
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2 | elioc1 9916 |
. . 3
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3 | 1, 2 | sylan2 286 |
. 2
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4 | mnfxr 8008 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
5 | 4 | a1i 9 |
. . . . . . 7
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6 | simpll 527 |
. . . . . . 7
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7 | simpr1 1003 |
. . . . . . 7
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8 | mnfle 9786 |
. . . . . . . 8
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9 | 8 | ad2antrr 488 |
. . . . . . 7
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10 | simpr2 1004 |
. . . . . . 7
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11 | 5, 6, 7, 9, 10 | xrlelttrd 9804 |
. . . . . 6
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12 | 1 | ad2antlr 489 |
. . . . . . 7
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13 | pnfxr 8004 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
14 | 13 | a1i 9 |
. . . . . . 7
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15 | simpr3 1005 |
. . . . . . 7
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16 | ltpnf 9774 |
. . . . . . . 8
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17 | 16 | ad2antlr 489 |
. . . . . . 7
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18 | 7, 12, 14, 15, 17 | xrlelttrd 9804 |
. . . . . 6
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19 | xrrebnd 9813 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 7, 19 | syl 14 |
. . . . . 6
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21 | 11, 18, 20 | mpbir2and 944 |
. . . . 5
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22 | 21, 10, 15 | 3jca 1177 |
. . . 4
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23 | 22 | ex 115 |
. . 3
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24 | rexr 7997 |
. . . 4
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25 | 24 | 3anim1i 1185 |
. . 3
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26 | 23, 25 | impbid1 142 |
. 2
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27 | 3, 26 | bitrd 188 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-cnex 7897 ax-resscn 7898 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-br 4002 df-opab 4063 df-id 4291 df-po 4294 df-iso 4295 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-iota 5175 df-fun 5215 df-fv 5221 df-ov 5873 df-oprab 5874 df-mpo 5875 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-ioc 9887 |
This theorem is referenced by: iocssre 9947 ef01bndlem 11755 sin01bnd 11756 cos01bnd 11757 cos1bnd 11758 sin01gt0 11760 cos01gt0 11761 sin02gt0 11762 sincos1sgn 11763 sincos2sgn 11764 cos12dec 11766 sin0pilem1 13984 sin0pilem2 13985 sinhalfpilem 13994 sincosq1lem 14028 coseq0negpitopi 14039 tangtx 14041 sincos4thpi 14043 pigt3 14047 |
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