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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 7980 |
. . 3
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2 | elioc1 9896 |
. . 3
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3 | 1, 2 | sylan2 286 |
. 2
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4 | mnfxr 7991 |
. . . . . . . 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 9766 |
. . . . . . . 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 9784 |
. . . . . 6
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12 | 1 | ad2antlr 489 |
. . . . . . 7
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13 | pnfxr 7987 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
14 | 13 | a1i 9 |
. . . . . . 7
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15 | simpr3 1005 |
. . . . . . 7
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16 | ltpnf 9754 |
. . . . . . . 8
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17 | 16 | ad2antlr 489 |
. . . . . . 7
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18 | 7, 12, 14, 15, 17 | xrlelttrd 9784 |
. . . . . 6
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19 | xrrebnd 9793 |
. . . . . . 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 7980 |
. . . 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 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-ioc 9867 |
This theorem is referenced by: iocssre 9927 ef01bndlem 11735 sin01bnd 11736 cos01bnd 11737 cos1bnd 11738 sin01gt0 11740 cos01gt0 11741 sin02gt0 11742 sincos1sgn 11743 sincos2sgn 11744 cos12dec 11746 sin0pilem1 13835 sin0pilem2 13836 sinhalfpilem 13845 sincosq1lem 13879 coseq0negpitopi 13890 tangtx 13892 sincos4thpi 13894 pigt3 13898 |
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