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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 9953 |
. . . 4
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2 | sstr 3163 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | ancoms 268 |
. . . 4
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4 | 1, 3 | sylan 283 |
. . 3
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5 | 4 | 3adant3 1017 |
. 2
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6 | ne0i 3429 |
. . 3
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7 | 6 | 3ad2ant3 1020 |
. 2
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8 | simplr 528 |
. . . 4
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9 | ssel 3149 |
. . . . . . . 8
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10 | elicc2 9936 |
. . . . . . . . 9
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11 | 10 | biimpd 144 |
. . . . . . . 8
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12 | 9, 11 | sylan9r 410 |
. . . . . . 7
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13 | 12 | imp 124 |
. . . . . 6
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14 | 13 | simp3d 1011 |
. . . . 5
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15 | 14 | ralrimiva 2550 |
. . . 4
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16 | breq2 4007 |
. . . . . 6
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17 | 16 | ralbidv 2477 |
. . . . 5
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18 | 17 | rspcev 2841 |
. . . 4
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19 | 8, 15, 18 | syl2anc 411 |
. . 3
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20 | 19 | 3adant3 1017 |
. 2
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21 | 5, 7, 20 | 3jca 1177 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 |
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-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-icc 9893 |
This theorem is referenced by: (None) |
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