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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 605 | . . . . . . . 8 | |
2 | 1 | expd 256 | . . . . . . 7 |
3 | dfnot 1353 | . . . . . . 7 | |
4 | 2, 3 | syl6ibr 161 | . . . . . 6 |
5 | 4 | com12 30 | . . . . 5 |
6 | 5 | imdistani 442 | . . . 4 |
7 | simpr 109 | . . . . . 6 | |
8 | 7 | con3i 622 | . . . . 5 |
9 | 8 | anim2i 340 | . . . 4 |
10 | 6, 9 | impbii 125 | . . 3 |
11 | eldif 3111 | . . . 4 | |
12 | elin 3291 | . . . . . 6 | |
13 | 12 | notbii 658 | . . . . 5 |
14 | 13 | anbi2i 453 | . . . 4 |
15 | 11, 14 | bitri 183 | . . 3 |
16 | eldif 3111 | . . 3 | |
17 | 10, 15, 16 | 3bitr4i 211 | . 2 |
18 | 17 | eqriv 2154 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1335 wfal 1340 wcel 2128 cdif 3099 cin 3101 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-in 3108 |
This theorem is referenced by: inssddif 3349 symdif1 3373 notrab 3385 disjdif2 3473 unfiin 6872 bj-charfundcALT 13455 |
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