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Mirrors > Home > ILE Home > Th. List > dcun | Unicode version |
Description: The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) |
Ref | Expression |
---|---|
dcun.a | DECID |
dcun.b | DECID |
Ref | Expression |
---|---|
dcun | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun1 3213 | . . . . 5 | |
2 | 1 | adantl 275 | . . . 4 |
3 | 2 | orcd 707 | . . 3 |
4 | df-dc 805 | . . 3 DECID | |
5 | 3, 4 | sylibr 133 | . 2 DECID |
6 | elun2 3214 | . . . . . 6 | |
7 | 6 | adantl 275 | . . . . 5 |
8 | 7 | orcd 707 | . . . 4 |
9 | 8, 4 | sylibr 133 | . . 3 DECID |
10 | simplr 504 | . . . . . . 7 | |
11 | simpr 109 | . . . . . . 7 | |
12 | ioran 726 | . . . . . . 7 | |
13 | 10, 11, 12 | sylanbrc 413 | . . . . . 6 |
14 | elun 3187 | . . . . . 6 | |
15 | 13, 14 | sylnibr 651 | . . . . 5 |
16 | 15 | olcd 708 | . . . 4 |
17 | 16, 4 | sylibr 133 | . . 3 DECID |
18 | dcun.b | . . . . 5 DECID | |
19 | exmiddc 806 | . . . . 5 DECID | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 20 | adantr 274 | . . 3 |
22 | 9, 17, 21 | mpjaodan 772 | . 2 DECID |
23 | dcun.a | . . 3 DECID | |
24 | exmiddc 806 | . . 3 DECID | |
25 | 23, 24 | syl 14 | . 2 |
26 | 5, 22, 25 | mpjaodan 772 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 DECID wdc 804 wcel 1465 cun 3039 |
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-v 2662 df-un 3045 df-in 3047 df-ss 3054 |
This theorem is referenced by: sumsplitdc 11156 |
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