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Mirrors > Home > ILE Home > Th. List > suplubti | Unicode version |
Description: A supremum is the least upper bound. See also supclti 6773 and supubti 6774. (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 109 |
. . . . . 6
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2 | breq1 3870 |
. . . . . . . 8
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3 | breq1 3870 |
. . . . . . . . 9
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4 | 3 | rexbidv 2392 |
. . . . . . . 8
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5 | 2, 4 | imbi12d 233 |
. . . . . . 7
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6 | 5 | cbvralv 2604 |
. . . . . 6
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7 | 1, 6 | sylib 121 |
. . . . 5
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8 | 7 | a1i 9 |
. . . 4
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9 | 8 | ss2rabi 3118 |
. . 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 6770 |
. . . 4
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13 | 10, 11 | supeuti 6769 |
. . . . 5
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14 | riotacl2 5659 |
. . . . 5
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15 | 13, 14 | syl 14 |
. . . 4
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16 | 12, 15 | eqeltrd 2171 |
. . 3
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17 | 9, 16 | sseldi 3037 |
. 2
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18 | breq2 3871 |
. . . . . 6
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19 | 18 | imbi1d 230 |
. . . . 5
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20 | 19 | ralbidv 2391 |
. . . 4
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21 | 20 | elrab 2785 |
. . 3
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22 | 21 | simprbi 270 |
. 2
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23 | breq1 3870 |
. . . . 5
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24 | breq1 3870 |
. . . . . 6
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25 | 24 | rexbidv 2392 |
. . . . 5
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26 | 23, 25 | imbi12d 233 |
. . . 4
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27 | 26 | rspccv 2733 |
. . 3
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28 | 27 | impd 252 |
. 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-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-iota 5014 df-riota 5646 df-sup 6759 |
This theorem is referenced by: suplub2ti 6776 supisoti 6785 infglbti 6800 sup3exmid 8515 maxleast 10777 |
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