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Mirrors > Home > ILE Home > Th. List > rabrsndc | Unicode version |
Description: A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.) |
Ref | Expression |
---|---|
rabrsndc.1 |
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rabrsndc.2 |
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Ref | Expression |
---|---|
rabrsndc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabrsndc.1 |
. . . . . 6
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2 | rabrsndc.2 |
. . . . . . . 8
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3 | pm2.1dc 837 |
. . . . . . . 8
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4 | 2, 3 | ax-mp 5 |
. . . . . . 7
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5 | 4 | sbcth 2976 |
. . . . . 6
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6 | 1, 5 | ax-mp 5 |
. . . . 5
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7 | sbcor 3007 |
. . . . 5
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8 | 6, 7 | mpbi 145 |
. . . 4
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9 | ralsns 3630 |
. . . . . 6
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10 | 1, 9 | ax-mp 5 |
. . . . 5
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11 | ralsns 3630 |
. . . . . 6
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12 | 1, 11 | ax-mp 5 |
. . . . 5
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13 | 10, 12 | orbi12i 764 |
. . . 4
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14 | 8, 13 | mpbir 146 |
. . 3
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15 | rabeq0 3452 |
. . . 4
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16 | eqcom 2179 |
. . . . 5
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17 | rabid2 2653 |
. . . . 5
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18 | 16, 17 | bitri 184 |
. . . 4
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19 | 15, 18 | orbi12i 764 |
. . 3
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20 | 14, 19 | mpbir 146 |
. 2
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21 | eqeq1 2184 |
. . 3
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22 | eqeq1 2184 |
. . 3
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23 | 21, 22 | orbi12d 793 |
. 2
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24 | 20, 23 | mpbiri 168 |
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-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-nul 3423 df-sn 3598 |
This theorem is referenced by: (None) |
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