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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 | STAB |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2151 | . 2 | |
2 | eldif 3111 | . . . . . . 7 | |
3 | vex 2715 | . . . . . . . 8 | |
4 | 3 | biantrur 301 | . . . . . . 7 |
5 | eldif 3111 | . . . . . . . . 9 | |
6 | 3 | biantrur 301 | . . . . . . . . 9 |
7 | 5, 6 | bitr4i 186 | . . . . . . . 8 |
8 | 7 | notbii 658 | . . . . . . 7 |
9 | 2, 4, 8 | 3bitr2i 207 | . . . . . 6 |
10 | 9 | bibi1i 227 | . . . . 5 |
11 | biimp 117 | . . . . . 6 | |
12 | id 19 | . . . . . . 7 | |
13 | notnot 619 | . . . . . . 7 | |
14 | 12, 13 | impbid1 141 | . . . . . 6 |
15 | 11, 14 | impbii 125 | . . . . 5 |
16 | 10, 15 | bitri 183 | . . . 4 |
17 | df-stab 817 | . . . 4 STAB | |
18 | 16, 17 | bitr4i 186 | . . 3 STAB |
19 | 18 | albii 1450 | . 2 STAB |
20 | 1, 19 | bitri 183 | 1 STAB |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 STAB wstab 816 wal 1333 wceq 1335 wcel 2128 cvv 2712 cdif 3099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 |
This theorem is referenced by: (None) |
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