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Mirrors > Home > ILE Home > Th. List > supubti | Unicode version |
Description: A supremum is an upper
bound. See also supclti 6893 and suplubti 6895.
This proof demonstrates how to expand an iota-based definition (df-iota 5096) using riotacl2 5751. (Contributed by Jim Kingdon, 24-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti |
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supclti.2 |
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Ref | Expression |
---|---|
supubti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 |
. . . . 5
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2 | 1 | a1i 9 |
. . . 4
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3 | 2 | ss2rabi 3184 |
. . 3
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4 | supmoti.ti |
. . . . 5
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5 | supclti.2 |
. . . . 5
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6 | 4, 5 | supval2ti 6890 |
. . . 4
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7 | 4, 5 | supeuti 6889 |
. . . . 5
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8 | riotacl2 5751 |
. . . . 5
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9 | 7, 8 | syl 14 |
. . . 4
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10 | 6, 9 | eqeltrd 2217 |
. . 3
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11 | 3, 10 | sseldi 3100 |
. 2
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12 | breq2 3941 |
. . . . . . 7
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13 | 12 | notbid 657 |
. . . . . 6
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14 | 13 | cbvralv 2657 |
. . . . 5
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15 | breq1 3940 |
. . . . . . 7
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16 | 15 | notbid 657 |
. . . . . 6
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17 | 16 | ralbidv 2438 |
. . . . 5
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18 | 14, 17 | syl5bb 191 |
. . . 4
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19 | 18 | elrab 2844 |
. . 3
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20 | 19 | simprbi 273 |
. 2
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21 | breq2 3941 |
. . . 4
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22 | 21 | notbid 657 |
. . 3
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23 | 22 | rspccv 2790 |
. 2
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24 | 11, 20, 23 | 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 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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 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-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-riota 5738 df-sup 6879 |
This theorem is referenced by: suplub2ti 6896 supisoti 6905 inflbti 6919 suprubex 8733 zsupcl 11676 dvdslegcd 11689 |
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