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Mirrors > Home > ILE Home > Th. List > difin | Unicode version |
Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
difin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-in2 577 | . . . . . . . 8 | |
2 | 1 | expd 254 | . . . . . . 7 |
3 | dfnot 1302 | . . . . . . 7 | |
4 | 2, 3 | syl6ibr 160 | . . . . . 6 |
5 | 4 | com12 30 | . . . . 5 |
6 | 5 | imdistani 433 | . . . 4 |
7 | simpr 108 | . . . . . 6 | |
8 | 7 | con3i 594 | . . . . 5 |
9 | 8 | anim2i 334 | . . . 4 |
10 | 6, 9 | impbii 124 | . . 3 |
11 | eldif 2982 | . . . 4 | |
12 | elin 3155 | . . . . . 6 | |
13 | 12 | notbii 626 | . . . . 5 |
14 | 13 | anbi2i 444 | . . . 4 |
15 | 11, 14 | bitri 182 | . . 3 |
16 | eldif 2982 | . . 3 | |
17 | 10, 15, 16 | 3bitr4i 210 | . 2 |
18 | 17 | eqriv 2078 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wceq 1284 wfal 1289 wcel 1433 cdif 2970 cin 2972 |
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-fal 1290 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 |
This theorem is referenced by: inssddif 3205 symdif1 3229 notrab 3241 |
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