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Mirrors > Home > ILE Home > Th. List > inssdif0im | Unicode version |
Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.) |
Ref | Expression |
---|---|
inssdif0im |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3155 | . . . . . 6 | |
2 | 1 | imbi1i 236 | . . . . 5 |
3 | imanim 818 | . . . . 5 | |
4 | 2, 3 | sylbi 119 | . . . 4 |
5 | eldif 2982 | . . . . . 6 | |
6 | 5 | anbi2i 444 | . . . . 5 |
7 | elin 3155 | . . . . 5 | |
8 | anass 393 | . . . . 5 | |
9 | 6, 7, 8 | 3bitr4ri 211 | . . . 4 |
10 | 4, 9 | sylnib 633 | . . 3 |
11 | 10 | alimi 1384 | . 2 |
12 | dfss2 2988 | . 2 | |
13 | eq0 3266 | . 2 | |
14 | 11, 12, 13 | 3imtr4i 199 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wal 1282 wceq 1284 wcel 1433 cdif 2970 cin 2972 wss 2973 c0 3251 |
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-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 df-ss 2986 df-nul 3252 |
This theorem is referenced by: disjdif 3316 |
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