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Mirrors > Home > ILE Home > Th. List > undifdcss | Unicode version |
Description: Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.) |
Ref | Expression |
---|---|
undifdcss |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 3063 |
. . . 4
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2 | undifss 3344 |
. . . 4
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3 | 1, 2 | sylibr 132 |
. . 3
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4 | eleq2 2146 |
. . . . . . . 8
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5 | 4 | biimpa 290 |
. . . . . . 7
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6 | elun 3125 |
. . . . . . 7
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7 | 5, 6 | sylib 120 |
. . . . . 6
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8 | eldifn 3107 |
. . . . . . 7
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9 | 8 | orim2i 711 |
. . . . . 6
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10 | 7, 9 | syl 14 |
. . . . 5
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11 | df-dc 777 |
. . . . 5
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12 | 10, 11 | sylibr 132 |
. . . 4
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13 | 12 | ralrimiva 2440 |
. . 3
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14 | 3, 13 | jca 300 |
. 2
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15 | elun1 3151 |
. . . . . . 7
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16 | 15 | adantl 271 |
. . . . . 6
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17 | simplr 497 |
. . . . . . . 8
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18 | simpr 108 |
. . . . . . . 8
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19 | 17, 18 | eldifd 2994 |
. . . . . . 7
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20 | elun2 3152 |
. . . . . . 7
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21 | 19, 20 | syl 14 |
. . . . . 6
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22 | eleq1 2145 |
. . . . . . . . 9
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23 | 22 | dcbid 782 |
. . . . . . . 8
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24 | simplr 497 |
. . . . . . . 8
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25 | simpr 108 |
. . . . . . . 8
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26 | 23, 24, 25 | rspcdva 2717 |
. . . . . . 7
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27 | exmiddc 778 |
. . . . . . 7
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28 | 26, 27 | syl 14 |
. . . . . 6
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29 | 16, 21, 28 | mpjaodan 745 |
. . . . 5
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30 | 29 | ex 113 |
. . . 4
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31 | 30 | ssrdv 3016 |
. . 3
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32 | 2 | biimpi 118 |
. . . 4
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33 | 32 | adantr 270 |
. . 3
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34 | 31, 33 | eqssd 3027 |
. 2
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35 | 14, 34 | impbii 124 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 |
This theorem is referenced by: exmidfodomrlemim 6728 |
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