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Mirrors > Home > ILE Home > Th. List > Mathboxes > decidin | Unicode version |
Description: If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.) |
Ref | Expression |
---|---|
decidin.ss |
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decidin.a |
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decidin.b |
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Ref | Expression |
---|---|
decidin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decidin.b |
. . . 4
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2 | decidi 11695 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | decidin.a |
. . . . 5
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5 | decidi 11695 |
. . . . 5
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6 | 4, 5 | syl 14 |
. . . 4
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7 | decidin.ss |
. . . . . 6
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8 | 7 | ssneld 3027 |
. . . . 5
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9 | olc 667 |
. . . . 5
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10 | 8, 9 | syl6 33 |
. . . 4
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11 | 6, 10 | jaod 672 |
. . 3
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12 | 3, 11 | syld 44 |
. 2
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13 | 12 | decidr 11696 |
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-ral 2364 df-v 2621 df-in 3005 df-ss 3012 df-dcin 11694 |
This theorem is referenced by: sumdc2 11699 |
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