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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 3102 |
. . . 4
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2 | undifss 3390 |
. . . 4
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3 | 1, 2 | sylibr 133 |
. . 3
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4 | eleq2 2163 |
. . . . . . . 8
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5 | 4 | biimpa 292 |
. . . . . . 7
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6 | elun 3164 |
. . . . . . 7
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7 | 5, 6 | sylib 121 |
. . . . . 6
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8 | eldifn 3146 |
. . . . . . 7
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9 | 8 | orim2i 719 |
. . . . . 6
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10 | 7, 9 | syl 14 |
. . . . 5
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11 | df-dc 787 |
. . . . 5
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12 | 10, 11 | sylibr 133 |
. . . 4
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13 | 12 | ralrimiva 2464 |
. . 3
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14 | 3, 13 | jca 302 |
. 2
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15 | elun1 3190 |
. . . . . . 7
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16 | 15 | adantl 273 |
. . . . . 6
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17 | simplr 500 |
. . . . . . . 8
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18 | simpr 109 |
. . . . . . . 8
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19 | 17, 18 | eldifd 3031 |
. . . . . . 7
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20 | elun2 3191 |
. . . . . . 7
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21 | 19, 20 | syl 14 |
. . . . . 6
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22 | eleq1 2162 |
. . . . . . . . 9
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23 | 22 | dcbid 792 |
. . . . . . . 8
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24 | simplr 500 |
. . . . . . . 8
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25 | simpr 109 |
. . . . . . . 8
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26 | 23, 24, 25 | rspcdva 2749 |
. . . . . . 7
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27 | exmiddc 788 |
. . . . . . 7
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28 | 26, 27 | syl 14 |
. . . . . 6
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29 | 16, 21, 28 | mpjaodan 753 |
. . . . 5
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30 | 29 | ex 114 |
. . . 4
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31 | 30 | ssrdv 3053 |
. . 3
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32 | 2 | biimpi 119 |
. . . 4
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33 | 32 | adantr 272 |
. . 3
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34 | 31, 33 | eqssd 3064 |
. 2
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35 | 14, 34 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 |
This theorem is referenced by: sbthlemi5 6777 sbthlemi6 6778 exmidfodomrlemim 6966 |
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