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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 3202 | . . . 4 | |
2 | undifss 3494 | . . . 4 | |
3 | 1, 2 | sylibr 133 | . . 3 |
4 | eleq2 2234 | . . . . . . . 8 | |
5 | 4 | biimpa 294 | . . . . . . 7 |
6 | elun 3268 | . . . . . . 7 | |
7 | 5, 6 | sylib 121 | . . . . . 6 |
8 | eldifn 3250 | . . . . . . 7 | |
9 | 8 | orim2i 756 | . . . . . 6 |
10 | 7, 9 | syl 14 | . . . . 5 |
11 | df-dc 830 | . . . . 5 DECID | |
12 | 10, 11 | sylibr 133 | . . . 4 DECID |
13 | 12 | ralrimiva 2543 | . . 3 DECID |
14 | 3, 13 | jca 304 | . 2 DECID |
15 | elun1 3294 | . . . . . . 7 | |
16 | 15 | adantl 275 | . . . . . 6 DECID |
17 | simplr 525 | . . . . . . . 8 DECID | |
18 | simpr 109 | . . . . . . . 8 DECID | |
19 | 17, 18 | eldifd 3131 | . . . . . . 7 DECID |
20 | elun2 3295 | . . . . . . 7 | |
21 | 19, 20 | syl 14 | . . . . . 6 DECID |
22 | eleq1 2233 | . . . . . . . . 9 | |
23 | 22 | dcbid 833 | . . . . . . . 8 DECID DECID |
24 | simplr 525 | . . . . . . . 8 DECID DECID | |
25 | simpr 109 | . . . . . . . 8 DECID | |
26 | 23, 24, 25 | rspcdva 2839 | . . . . . . 7 DECID DECID |
27 | exmiddc 831 | . . . . . . 7 DECID | |
28 | 26, 27 | syl 14 | . . . . . 6 DECID |
29 | 16, 21, 28 | mpjaodan 793 | . . . . 5 DECID |
30 | 29 | ex 114 | . . . 4 DECID |
31 | 30 | ssrdv 3153 | . . 3 DECID |
32 | 2 | biimpi 119 | . . . 4 |
33 | 32 | adantr 274 | . . 3 DECID |
34 | 31, 33 | eqssd 3164 | . 2 DECID |
35 | 14, 34 | impbii 125 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 cdif 3118 cun 3119 wss 3121 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 |
This theorem is referenced by: sbthlemi5 6935 sbthlemi6 6936 exmidfodomrlemim 7167 bj-charfundcALT 13806 |
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