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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 |
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dcun.b |
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Ref | Expression |
---|---|
dcun |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun1 3167 |
. . . . 5
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2 | 1 | adantl 271 |
. . . 4
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3 | 2 | orcd 687 |
. . 3
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4 | df-dc 781 |
. . 3
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5 | 3, 4 | sylibr 132 |
. 2
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6 | elun2 3168 |
. . . . . 6
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7 | 6 | adantl 271 |
. . . . 5
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8 | 7 | orcd 687 |
. . . 4
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9 | 8, 4 | sylibr 132 |
. . 3
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10 | simplr 497 |
. . . . . . 7
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11 | simpr 108 |
. . . . . . 7
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12 | ioran 704 |
. . . . . . 7
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13 | 10, 11, 12 | sylanbrc 408 |
. . . . . 6
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14 | elun 3141 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | sylnibr 637 |
. . . . 5
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16 | 15 | olcd 688 |
. . . 4
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17 | 16, 4 | sylibr 132 |
. . 3
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18 | dcun.b |
. . . . 5
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19 | exmiddc 782 |
. . . . 5
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20 | 18, 19 | syl 14 |
. . . 4
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21 | 20 | adantr 270 |
. . 3
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22 | 9, 17, 21 | mpjaodan 747 |
. 2
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23 | dcun.a |
. . 3
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24 | exmiddc 782 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | syl 14 |
. 2
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26 | 5, 22, 25 | mpjaodan 747 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 |
This theorem is referenced by: sumsplitdc 10813 |
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