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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 545 | . . . . . . . 8 | |
2 | 1 | expd 245 | . . . . . . 7 |
3 | dfnot 1262 | . . . . . . 7 | |
4 | 2, 3 | syl6ibr 151 | . . . . . 6 |
5 | 4 | com12 27 | . . . . 5 |
6 | 5 | imdistani 419 | . . . 4 |
7 | simpr 103 | . . . . . 6 | |
8 | 7 | con3i 562 | . . . . 5 |
9 | 8 | anim2i 324 | . . . 4 |
10 | 6, 9 | impbii 117 | . . 3 |
11 | eldif 2927 | . . . 4 | |
12 | elin 3126 | . . . . . 6 | |
13 | 12 | notbii 594 | . . . . 5 |
14 | 13 | anbi2i 430 | . . . 4 |
15 | 11, 14 | bitri 173 | . . 3 |
16 | eldif 2927 | . . 3 | |
17 | 10, 15, 16 | 3bitr4i 201 | . 2 |
18 | 17 | eqriv 2037 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wceq 1243 wfal 1248 wcel 1393 cdif 2914 cin 2916 |
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-fal 1249 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 |
This theorem is referenced by: inssddif 3178 symdif1 3202 notrab 3214 |
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