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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 | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 3122 | . . . 4 | |
2 | undifss 3413 | . . . 4 | |
3 | 1, 2 | sylibr 133 | . . 3 |
4 | eleq2 2181 | . . . . . . . 8 | |
5 | 4 | biimpa 294 | . . . . . . 7 |
6 | elun 3187 | . . . . . . 7 | |
7 | 5, 6 | sylib 121 | . . . . . 6 |
8 | eldifn 3169 | . . . . . . 7 | |
9 | 8 | orim2i 735 | . . . . . 6 |
10 | 7, 9 | syl 14 | . . . . 5 |
11 | df-dc 805 | . . . . 5 DECID | |
12 | 10, 11 | sylibr 133 | . . . 4 DECID |
13 | 12 | ralrimiva 2482 | . . 3 DECID |
14 | 3, 13 | jca 304 | . 2 DECID |
15 | elun1 3213 | . . . . . . 7 | |
16 | 15 | adantl 275 | . . . . . 6 DECID |
17 | simplr 504 | . . . . . . . 8 DECID | |
18 | simpr 109 | . . . . . . . 8 DECID | |
19 | 17, 18 | eldifd 3051 | . . . . . . 7 DECID |
20 | elun2 3214 | . . . . . . 7 | |
21 | 19, 20 | syl 14 | . . . . . 6 DECID |
22 | eleq1 2180 | . . . . . . . . 9 | |
23 | 22 | dcbid 808 | . . . . . . . 8 DECID DECID |
24 | simplr 504 | . . . . . . . 8 DECID DECID | |
25 | simpr 109 | . . . . . . . 8 DECID | |
26 | 23, 24, 25 | rspcdva 2768 | . . . . . . 7 DECID DECID |
27 | exmiddc 806 | . . . . . . 7 DECID | |
28 | 26, 27 | syl 14 | . . . . . 6 DECID |
29 | 16, 21, 28 | mpjaodan 772 | . . . . 5 DECID |
30 | 29 | ex 114 | . . . 4 DECID |
31 | 30 | ssrdv 3073 | . . 3 DECID |
32 | 2 | biimpi 119 | . . . 4 |
33 | 32 | adantr 274 | . . 3 DECID |
34 | 31, 33 | eqssd 3084 | . 2 DECID |
35 | 14, 34 | impbii 125 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 682 DECID wdc 804 wceq 1316 wcel 1465 wral 2393 cdif 3038 cun 3039 wss 3041 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 |
This theorem is referenced by: sbthlemi5 6817 sbthlemi6 6818 exmidfodomrlemim 7025 |
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