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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 3302 |
. . . . 5
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2 | 1 | adantl 277 |
. . . 4
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3 | 2 | orcd 733 |
. . 3
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4 | df-dc 835 |
. . 3
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5 | 3, 4 | sylibr 134 |
. 2
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6 | elun2 3303 |
. . . . . 6
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7 | 6 | adantl 277 |
. . . . 5
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8 | 7 | orcd 733 |
. . . 4
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9 | 8, 4 | sylibr 134 |
. . 3
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10 | simplr 528 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | simpr 110 |
. . . . . . 7
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12 | ioran 752 |
. . . . . . 7
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13 | 10, 11, 12 | sylanbrc 417 |
. . . . . 6
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14 | elun 3276 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | sylnibr 677 |
. . . . 5
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16 | 15 | olcd 734 |
. . . 4
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17 | 16, 4 | sylibr 134 |
. . 3
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18 | dcun.b |
. . . . 5
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19 | exmiddc 836 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | syl 14 |
. . . 4
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21 | 20 | adantr 276 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 9, 17, 21 | mpjaodan 798 |
. 2
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23 | dcun.a |
. . 3
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24 | exmiddc 836 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | syl 14 |
. 2
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26 | 5, 22, 25 | mpjaodan 798 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 |
This theorem is referenced by: sumsplitdc 11435 |
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