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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 610 | . . . . . . . 8 | |
2 | 1 | expd 256 | . . . . . . 7 |
3 | dfnot 1366 | . . . . . . 7 | |
4 | 2, 3 | syl6ibr 161 | . . . . . 6 |
5 | 4 | com12 30 | . . . . 5 |
6 | 5 | imdistani 443 | . . . 4 |
7 | simpr 109 | . . . . . 6 | |
8 | 7 | con3i 627 | . . . . 5 |
9 | 8 | anim2i 340 | . . . 4 |
10 | 6, 9 | impbii 125 | . . 3 |
11 | eldif 3130 | . . . 4 | |
12 | elin 3310 | . . . . . 6 | |
13 | 12 | notbii 663 | . . . . 5 |
14 | 13 | anbi2i 454 | . . . 4 |
15 | 11, 14 | bitri 183 | . . 3 |
16 | eldif 3130 | . . 3 | |
17 | 10, 15, 16 | 3bitr4i 211 | . 2 |
18 | 17 | eqriv 2167 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1348 wfal 1353 wcel 2141 cdif 3118 cin 3120 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 |
This theorem is referenced by: inssddif 3368 symdif1 3392 notrab 3404 disjdif2 3493 unfiin 6903 bj-charfundcALT 13844 |
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