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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 3258 | . . 3 | |
2 | eldif 3120 | . . . . . . 7 | |
3 | eldif 3120 | . . . . . . 7 | |
4 | 2, 3 | orbi12i 754 | . . . . . 6 |
5 | andi 808 | . . . . . 6 | |
6 | 4, 5 | bitr4i 186 | . . . . 5 |
7 | pm3.14 743 | . . . . . 6 | |
8 | 7 | anim2i 340 | . . . . 5 |
9 | 6, 8 | sylbi 120 | . . . 4 |
10 | eldif 3120 | . . . . 5 | |
11 | elin 3300 | . . . . . . 7 | |
12 | 11 | notbii 658 | . . . . . 6 |
13 | 12 | anbi2i 453 | . . . . 5 |
14 | 10, 13 | bitr2i 184 | . . . 4 |
15 | 9, 14 | sylib 121 | . . 3 |
16 | 1, 15 | sylbi 120 | . 2 |
17 | 16 | ssriv 3141 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 698 wcel 2135 cdif 3108 cun 3109 cin 3110 wss 3111 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 |
This theorem is referenced by: difdif2ss 3374 indmss 3376 |
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