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Mirrors > Home > ILE Home > Th. List > eqinfti | Unicode version |
Description: Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
Ref | Expression |
---|---|
eqinfti.ti |
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Ref | Expression |
---|---|
eqinfti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6880 |
. . 3
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2 | eqinfti.ti |
. . . . . 6
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3 | 2 | cnvti 6914 |
. . . . 5
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4 | 3 | eqsupti 6891 |
. . . 4
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5 | vex 2692 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() | |
6 | brcnvg 4728 |
. . . . . . . . . . . 12
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7 | 6 | bicomd 140 |
. . . . . . . . . . 11
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8 | 5, 7 | mpan2 422 |
. . . . . . . . . 10
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9 | 8 | notbid 657 |
. . . . . . . . 9
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10 | 9 | ralbidv 2438 |
. . . . . . . 8
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11 | brcnvg 4728 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 5, 11 | mpan 421 |
. . . . . . . . . . 11
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13 | 12 | bicomd 140 |
. . . . . . . . . 10
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14 | vex 2692 |
. . . . . . . . . . . . . 14
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15 | 5, 14 | brcnv 4730 |
. . . . . . . . . . . . 13
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16 | 15 | a1i 9 |
. . . . . . . . . . . 12
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17 | 16 | bicomd 140 |
. . . . . . . . . . 11
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18 | 17 | rexbidv 2439 |
. . . . . . . . . 10
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19 | 13, 18 | imbi12d 233 |
. . . . . . . . 9
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20 | 19 | ralbidv 2438 |
. . . . . . . 8
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21 | 10, 20 | anbi12d 465 |
. . . . . . 7
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22 | 21 | pm5.32i 450 |
. . . . . 6
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23 | 3anass 967 |
. . . . . 6
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24 | 3anass 967 |
. . . . . 6
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25 | 22, 23, 24 | 3bitr4i 211 |
. . . . 5
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26 | 25 | biimpi 119 |
. . . 4
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27 | 4, 26 | impel 278 |
. . 3
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28 | 1, 27 | syl5eq 2185 |
. 2
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29 | 28 | ex 114 |
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-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
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-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-cnv 4555 df-iota 5096 df-riota 5738 df-sup 6879 df-inf 6880 |
This theorem is referenced by: eqinftid 6916 |
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