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Mirrors > Home > ILE Home > Th. List > iccsupr | Unicode version |
Description: A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.) |
Ref | Expression |
---|---|
iccsupr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssre 9768 |
. . . 4
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2 | sstr 3110 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | ancoms 266 |
. . . 4
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4 | 1, 3 | sylan 281 |
. . 3
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5 | 4 | 3adant3 1002 |
. 2
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6 | ne0i 3374 |
. . 3
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7 | 6 | 3ad2ant3 1005 |
. 2
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8 | simplr 520 |
. . . 4
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9 | ssel 3096 |
. . . . . . . 8
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10 | elicc2 9751 |
. . . . . . . . 9
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11 | 10 | biimpd 143 |
. . . . . . . 8
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12 | 9, 11 | sylan9r 408 |
. . . . . . 7
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13 | 12 | imp 123 |
. . . . . 6
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14 | 13 | simp3d 996 |
. . . . 5
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15 | 14 | ralrimiva 2508 |
. . . 4
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16 | breq2 3941 |
. . . . . 6
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17 | 16 | ralbidv 2438 |
. . . . 5
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18 | 17 | rspcev 2793 |
. . . 4
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19 | 8, 15, 18 | syl2anc 409 |
. . 3
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20 | 19 | 3adant3 1002 |
. 2
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21 | 5, 7, 20 | 3jca 1162 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-icc 9708 |
This theorem is referenced by: (None) |
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