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Mirrors > Home > ILE Home > Th. List > ssddif | Unicode version |
Description: Double complement and subset. Similar to ddifss 3202 but inside a class instead of the universal class . In classical logic the subset operation on the right hand side could be an equality (that is, ). (Contributed by Jim Kingdon, 24-Jul-2018.) |
Ref | Expression |
---|---|
ssddif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancr 314 | . . . . 5 | |
2 | simpr 108 | . . . . . . . 8 | |
3 | 2 | con2i 589 | . . . . . . 7 |
4 | 3 | anim2i 334 | . . . . . 6 |
5 | eldif 2982 | . . . . . . 7 | |
6 | eldif 2982 | . . . . . . . . 9 | |
7 | 6 | notbii 626 | . . . . . . . 8 |
8 | 7 | anbi2i 444 | . . . . . . 7 |
9 | 5, 8 | bitri 182 | . . . . . 6 |
10 | 4, 9 | sylibr 132 | . . . . 5 |
11 | 1, 10 | syl6 33 | . . . 4 |
12 | eldifi 3094 | . . . . 5 | |
13 | 12 | imim2i 12 | . . . 4 |
14 | 11, 13 | impbii 124 | . . 3 |
15 | 14 | albii 1399 | . 2 |
16 | dfss2 2988 | . 2 | |
17 | dfss2 2988 | . 2 | |
18 | 15, 16, 17 | 3bitr4i 210 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wal 1282 wcel 1433 cdif 2970 wss 2973 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 |
This theorem is referenced by: ddifss 3202 inssddif 3205 |
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