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Mirrors > Home > ILE Home > Th. List > supubti | Unicode version |
Description: A supremum is an upper
bound. See also supclti 7031 and suplubti 7033.
This proof demonstrates how to expand an iota-based definition (df-iota 5199) using riotacl2 5869. (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 109 |
. . . . 5
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2 | 1 | a1i 9 |
. . . 4
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3 | 2 | ss2rabi 3252 |
. . 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 7028 |
. . . 4
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7 | 4, 5 | supeuti 7027 |
. . . . 5
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8 | riotacl2 5869 |
. . . . 5
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9 | 7, 8 | syl 14 |
. . . 4
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10 | 6, 9 | eqeltrd 2266 |
. . 3
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11 | 3, 10 | sselid 3168 |
. 2
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12 | breq2 4025 |
. . . . . . 7
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13 | 12 | notbid 668 |
. . . . . 6
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14 | 13 | cbvralv 2718 |
. . . . 5
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15 | breq1 4024 |
. . . . . . 7
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16 | 15 | notbid 668 |
. . . . . 6
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17 | 16 | ralbidv 2490 |
. . . . 5
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18 | 14, 17 | bitrid 192 |
. . . 4
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19 | 18 | elrab 2908 |
. . 3
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20 | 19 | simprbi 275 |
. 2
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21 | breq2 4025 |
. . . 4
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22 | 21 | notbid 668 |
. . 3
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23 | 22 | rspccv 2853 |
. 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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-iota 5199 df-riota 5855 df-sup 7017 |
This theorem is referenced by: suplub2ti 7034 supisoti 7043 inflbti 7057 suprubex 8943 zsupcl 11989 dvdslegcd 12006 |
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