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Mirrors > Home > ILE Home > Th. List > difindiss | Unicode version |
Description: Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.) |
Ref | Expression |
---|---|
difindiss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3084 | . . 3 | |
2 | eldif 2927 | . . . . . . 7 | |
3 | eldif 2927 | . . . . . . 7 | |
4 | 2, 3 | orbi12i 681 | . . . . . 6 |
5 | andi 731 | . . . . . 6 | |
6 | 4, 5 | bitr4i 176 | . . . . 5 |
7 | pm3.14 670 | . . . . . 6 | |
8 | 7 | anim2i 324 | . . . . 5 |
9 | 6, 8 | sylbi 114 | . . . 4 |
10 | eldif 2927 | . . . . 5 | |
11 | elin 3126 | . . . . . . 7 | |
12 | 11 | notbii 594 | . . . . . 6 |
13 | 12 | anbi2i 430 | . . . . 5 |
14 | 10, 13 | bitr2i 174 | . . . 4 |
15 | 9, 14 | sylib 127 | . . 3 |
16 | 1, 15 | sylbi 114 | . 2 |
17 | 16 | ssriv 2949 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 97 wo 629 wcel 1393 cdif 2914 cun 2915 cin 2916 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-un 2922 df-in 2924 df-ss 2931 |
This theorem is referenced by: difdif2ss 3194 indmss 3196 |
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