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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 3209 |
. . . . 5
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2 | 1 | adantl 273 |
. . . 4
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3 | 2 | orcd 705 |
. . 3
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4 | df-dc 803 |
. . 3
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5 | 3, 4 | sylibr 133 |
. 2
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6 | elun2 3210 |
. . . . . 6
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7 | 6 | adantl 273 |
. . . . 5
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8 | 7 | orcd 705 |
. . . 4
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9 | 8, 4 | sylibr 133 |
. . 3
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10 | simplr 502 |
. . . . . . 7
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11 | simpr 109 |
. . . . . . 7
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12 | ioran 724 |
. . . . . . 7
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13 | 10, 11, 12 | sylanbrc 411 |
. . . . . 6
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14 | elun 3183 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | sylnibr 649 |
. . . . 5
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16 | 15 | olcd 706 |
. . . 4
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17 | 16, 4 | sylibr 133 |
. . 3
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18 | dcun.b |
. . . . 5
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19 | exmiddc 804 |
. . . . 5
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20 | 18, 19 | syl 14 |
. . . 4
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21 | 20 | adantr 272 |
. . 3
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22 | 9, 17, 21 | mpjaodan 770 |
. 2
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23 | dcun.a |
. . 3
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24 | exmiddc 804 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | syl 14 |
. 2
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26 | 5, 22, 25 | mpjaodan 770 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 |
This theorem is referenced by: sumsplitdc 11093 |
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