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Mirrors > Home > ILE Home > Th. List > ddifstab | Unicode version |
Description: A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.) |
Ref | Expression |
---|---|
ddifstab |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2077 |
. 2
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2 | eldif 2993 |
. . . . . . 7
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3 | vex 2615 |
. . . . . . . 8
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4 | 3 | biantrur 297 |
. . . . . . 7
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5 | eldif 2993 |
. . . . . . . . 9
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6 | 3 | biantrur 297 |
. . . . . . . . 9
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7 | 5, 6 | bitr4i 185 |
. . . . . . . 8
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8 | 7 | notbii 627 |
. . . . . . 7
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9 | 2, 4, 8 | 3bitr2i 206 |
. . . . . 6
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10 | 9 | bibi1i 226 |
. . . . 5
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11 | bi1 116 |
. . . . . 6
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12 | id 19 |
. . . . . . 7
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13 | notnot 592 |
. . . . . . 7
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14 | 12, 13 | impbid1 140 |
. . . . . 6
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15 | 11, 14 | impbii 124 |
. . . . 5
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16 | 10, 15 | bitri 182 |
. . . 4
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17 | df-stab 774 |
. . . 4
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18 | 16, 17 | bitr4i 185 |
. . 3
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19 | 18 | albii 1400 |
. 2
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20 | 1, 19 | bitri 182 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-stab 774 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-dif 2986 |
This theorem is referenced by: (None) |
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