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Mirrors > Home > ILE Home > Th. List > suplubti | Unicode version |
Description: A supremum is the least upper bound. See also supclti 7010 and supubti 7011. (Contributed by Jim Kingdon, 24-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti |
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supclti.2 |
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Ref | Expression |
---|---|
suplubti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 |
. . . . . 6
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2 | breq1 4018 |
. . . . . . . 8
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3 | breq1 4018 |
. . . . . . . . 9
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4 | 3 | rexbidv 2488 |
. . . . . . . 8
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5 | 2, 4 | imbi12d 234 |
. . . . . . 7
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6 | 5 | cbvralv 2715 |
. . . . . 6
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7 | 1, 6 | sylib 122 |
. . . . 5
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8 | 7 | a1i 9 |
. . . 4
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9 | 8 | ss2rabi 3249 |
. . 3
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10 | supmoti.ti |
. . . . 5
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11 | supclti.2 |
. . . . 5
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12 | 10, 11 | supval2ti 7007 |
. . . 4
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13 | 10, 11 | supeuti 7006 |
. . . . 5
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14 | riotacl2 5857 |
. . . . 5
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15 | 13, 14 | syl 14 |
. . . 4
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16 | 12, 15 | eqeltrd 2264 |
. . 3
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17 | 9, 16 | sselid 3165 |
. 2
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18 | breq2 4019 |
. . . . . 6
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19 | 18 | imbi1d 231 |
. . . . 5
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20 | 19 | ralbidv 2487 |
. . . 4
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21 | 20 | elrab 2905 |
. . 3
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22 | 21 | simprbi 275 |
. 2
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23 | breq1 4018 |
. . . . 5
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24 | breq1 4018 |
. . . . . 6
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25 | 24 | rexbidv 2488 |
. . . . 5
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26 | 23, 25 | imbi12d 234 |
. . . 4
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27 | 26 | rspccv 2850 |
. . 3
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28 | 27 | impd 254 |
. 2
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29 | 17, 22, 28 | 3syl 17 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-iota 5190 df-riota 5844 df-sup 6996 |
This theorem is referenced by: suplub2ti 7013 supisoti 7022 infglbti 7037 sup3exmid 8927 maxleast 11235 |
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