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Mirrors > Home > ILE Home > Th. List > ssddif | Unicode version |
Description: Double complement and subset. Similar to ddifss 3175 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 304 | . . . . 5 | |
2 | simpr 103 | . . . . . . . 8 | |
3 | 2 | con2i 557 | . . . . . . 7 |
4 | 3 | anim2i 324 | . . . . . 6 |
5 | eldif 2927 | . . . . . . 7 | |
6 | eldif 2927 | . . . . . . . . 9 | |
7 | 6 | notbii 594 | . . . . . . . 8 |
8 | 7 | anbi2i 430 | . . . . . . 7 |
9 | 5, 8 | bitri 173 | . . . . . 6 |
10 | 4, 9 | sylibr 137 | . . . . 5 |
11 | 1, 10 | syl6 29 | . . . 4 |
12 | eldifi 3066 | . . . . 5 | |
13 | 12 | imim2i 12 | . . . 4 |
14 | 11, 13 | impbii 117 | . . 3 |
15 | 14 | albii 1359 | . 2 |
16 | dfss2 2934 | . 2 | |
17 | dfss2 2934 | . 2 | |
18 | 15, 16, 17 | 3bitr4i 201 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wal 1241 wcel 1393 cdif 2914 wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 |
This theorem is referenced by: ddifss 3175 inssddif 3178 |
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