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Mirrors > Home > MPE Home > Th. List > elico2 | Structured version Visualization version 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 9686 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | elico1 11679 |
. . 3
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3 | 1, 2 | sylan 474 |
. 2
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4 | mnfxr 11414 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
5 | 4 | a1i 11 |
. . . . . . 7
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6 | 1 | ad2antrr 732 |
. . . . . . 7
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7 | simpr1 1014 |
. . . . . . 7
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8 | mnflt 11425 |
. . . . . . . 8
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9 | 8 | ad2antrr 732 |
. . . . . . 7
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10 | simpr2 1015 |
. . . . . . 7
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11 | 5, 6, 7, 9, 10 | xrltletrd 11458 |
. . . . . 6
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12 | simplr 762 |
. . . . . . 7
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13 | pnfxr 11412 |
. . . . . . . 8
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14 | 13 | a1i 11 |
. . . . . . 7
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15 | simpr3 1016 |
. . . . . . 7
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16 | pnfge 11432 |
. . . . . . . 8
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17 | 16 | ad2antlr 733 |
. . . . . . 7
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18 | 7, 12, 14, 15, 17 | xrltletrd 11458 |
. . . . . 6
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19 | xrrebnd 11463 |
. . . . . . 7
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20 | 7, 19 | syl 17 |
. . . . . 6
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21 | 11, 18, 20 | mpbir2and 933 |
. . . . 5
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22 | 21, 10, 15 | 3jca 1188 |
. . . 4
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23 | 22 | ex 436 |
. . 3
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24 | rexr 9686 |
. . . 4
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25 | 24 | 3anim1i 1194 |
. . 3
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26 | 23, 25 | impbid1 207 |
. 2
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27 | 3, 26 | bitrd 257 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-pre-lttri 9613 ax-pre-lttrn 9614 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-po 4755 df-so 4756 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-ico 11641 |
This theorem is referenced by: icossre 11715 elicopnf 11730 icoshft 11754 muladdmodid 12137 icodiamlt 13497 fprodge0 14047 fprodge1 14049 rge0srg 19038 metustexhalf 21571 cnbl0 21794 icoopnst 21967 iocopnst 21968 icopnfcnv 21970 icopnfhmeo 21971 iccpnfcnv 21972 psercnlem2 23379 psercnlem1 23380 psercn 23381 abelth 23396 tanord1 23486 tanord 23487 efopnlem1 23601 logtayl 23605 rlimcnp 23891 rlimcnp2 23892 dchrvmasumlem2 24336 dchrvmasumiflem1 24339 pntlemb 24435 pnt 24452 ubico 28357 xrge0slmod 28607 voliune 29052 volfiniune 29053 dya2icoseg 29099 sibfinima 29172 relowlpssretop 31767 tan2h 31937 itg2addnclem2 31994 modelico 35665 binomcxplemdvbinom 36702 binomcxplemcvg 36703 binomcxplemnotnn0 36705 limciccioolb 37701 fourierdlem32 38002 fourierdlem43 38014 fourierdlem63 38033 fourierdlem79 38049 fouriersw 38095 expnegico01 40368 dignnld 40467 |
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