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Mirrors > Home > MPE Home > Th. List > iccss2 | Structured version Visualization version Unicode version |
Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
iccss2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-icc 11670 |
. . . . . 6
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2 | 1 | elixx3g 11676 |
. . . . 5
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3 | 2 | simplbi 466 |
. . . 4
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4 | 3 | adantr 471 |
. . 3
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5 | 4 | simp1d 1026 |
. 2
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6 | 4 | simp2d 1027 |
. 2
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7 | 2 | simprbi 470 |
. . . 4
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8 | 7 | adantr 471 |
. . 3
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9 | 8 | simpld 465 |
. 2
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10 | 1 | elixx3g 11676 |
. . . . 5
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11 | 10 | simprbi 470 |
. . . 4
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12 | 11 | simprd 469 |
. . 3
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13 | 12 | adantl 472 |
. 2
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14 | xrletr 11483 |
. . 3
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15 | xrletr 11483 |
. . 3
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16 | 1, 1, 14, 15 | ixxss12 11683 |
. 2
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17 | 5, 6, 9, 13, 16 | syl22anc 1277 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-cnex 9620 ax-resscn 9621 ax-pre-lttri 9638 ax-pre-lttrn 9639 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-op 3986 df-uni 4212 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-id 4767 df-po 4773 df-so 4774 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-1st 6819 df-2nd 6820 df-er 7388 df-en 7595 df-dom 7596 df-sdom 7597 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-icc 11670 |
This theorem is referenced by: ordtresticc 20287 iccconn 21896 icccvx 22026 oprpiece1res1 22027 oprpiece1res2 22028 pcoass 22103 dvlip 22993 c1liplem1 22996 dvgt0lem1 23002 ftc2ditglem 23045 ttgcontlem1 24963 unitssxrge0 28754 xrge0iifhmeo 28790 |
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