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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 3225 |
. . . 4
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2 | undifss 3518 |
. . . 4
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3 | 1, 2 | sylibr 134 |
. . 3
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4 | eleq2 2253 |
. . . . . . . 8
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5 | 4 | biimpa 296 |
. . . . . . 7
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6 | elun 3291 |
. . . . . . 7
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7 | 5, 6 | sylib 122 |
. . . . . 6
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8 | eldifn 3273 |
. . . . . . 7
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9 | 8 | orim2i 762 |
. . . . . 6
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10 | 7, 9 | syl 14 |
. . . . 5
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11 | df-dc 836 |
. . . . 5
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12 | 10, 11 | sylibr 134 |
. . . 4
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13 | 12 | ralrimiva 2563 |
. . 3
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14 | 3, 13 | jca 306 |
. 2
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15 | elun1 3317 |
. . . . . . 7
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16 | 15 | adantl 277 |
. . . . . 6
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17 | simplr 528 |
. . . . . . . 8
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18 | simpr 110 |
. . . . . . . 8
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19 | 17, 18 | eldifd 3154 |
. . . . . . 7
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20 | elun2 3318 |
. . . . . . 7
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21 | 19, 20 | syl 14 |
. . . . . 6
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22 | eleq1 2252 |
. . . . . . . . 9
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23 | 22 | dcbid 839 |
. . . . . . . 8
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24 | simplr 528 |
. . . . . . . 8
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25 | simpr 110 |
. . . . . . . 8
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26 | 23, 24, 25 | rspcdva 2861 |
. . . . . . 7
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27 | exmiddc 837 |
. . . . . . 7
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28 | 26, 27 | syl 14 |
. . . . . 6
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29 | 16, 21, 28 | mpjaodan 799 |
. . . . 5
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30 | 29 | ex 115 |
. . . 4
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31 | 30 | ssrdv 3176 |
. . 3
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32 | 2 | biimpi 120 |
. . . 4
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33 | 32 | adantr 276 |
. . 3
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34 | 31, 33 | eqssd 3187 |
. 2
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35 | 14, 34 | impbii 126 |
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-dc 836 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 |
This theorem is referenced by: sbthlemi5 6979 sbthlemi6 6980 exmidfodomrlemim 7219 bj-charfundcALT 14964 |
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