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Mirrors > Home > ILE Home > Th. List > elico2 | Unicode version |
Description: Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
elico2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7071 |
. . 3
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2 | elico1 8792 |
. . 3
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3 | 1, 2 | sylan 267 |
. 2
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4 | mnfxr 8694 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
5 | 4 | a1i 9 |
. . . . . . 7
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6 | 1 | ad2antrr 457 |
. . . . . . 7
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7 | simpr1 910 |
. . . . . . 7
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8 | mnflt 8704 |
. . . . . . . 8
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9 | 8 | ad2antrr 457 |
. . . . . . 7
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10 | simpr2 911 |
. . . . . . 7
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11 | 5, 6, 7, 9, 10 | xrltletrd 8727 |
. . . . . 6
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12 | simplr 482 |
. . . . . . 7
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13 | pnfxr 8692 |
. . . . . . . 8
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14 | 13 | a1i 9 |
. . . . . . 7
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15 | simpr3 912 |
. . . . . . 7
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16 | pnfge 8710 |
. . . . . . . 8
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17 | 16 | ad2antlr 458 |
. . . . . . 7
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18 | 7, 12, 14, 15, 17 | xrltletrd 8727 |
. . . . . 6
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19 | xrrebnd 8732 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 7, 19 | syl 14 |
. . . . . 6
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21 | 11, 18, 20 | mpbir2and 851 |
. . . . 5
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22 | 21, 10, 15 | 3jca 1084 |
. . . 4
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23 | 22 | ex 108 |
. . 3
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24 | rexr 7071 |
. . . 4
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25 | 24 | 3anim1i 1090 |
. . 3
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26 | 23, 25 | impbid1 130 |
. 2
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27 | 3, 26 | bitrd 177 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-ico 8763 |
This theorem is referenced by: icossre 8823 elicopnf 8838 icoshft 8858 modqelico 9176 icodiamlt 9776 |
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