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Mirrors > Home > ILE Home > Th. List > ssconb | Unicode version |
Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
ssconb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2993 | . . . . . . 7 | |
2 | ssel 2993 | . . . . . . 7 | |
3 | pm5.1 565 | . . . . . . 7 | |
4 | 1, 2, 3 | syl2an 283 | . . . . . 6 |
5 | con2b 625 | . . . . . . 7 | |
6 | 5 | a1i 9 | . . . . . 6 |
7 | 4, 6 | anbi12d 456 | . . . . 5 |
8 | jcab 567 | . . . . 5 | |
9 | jcab 567 | . . . . 5 | |
10 | 7, 8, 9 | 3bitr4g 221 | . . . 4 |
11 | eldif 2982 | . . . . 5 | |
12 | 11 | imbi2i 224 | . . . 4 |
13 | eldif 2982 | . . . . 5 | |
14 | 13 | imbi2i 224 | . . . 4 |
15 | 10, 12, 14 | 3bitr4g 221 | . . 3 |
16 | 15 | albidv 1745 | . 2 |
17 | dfss2 2988 | . 2 | |
18 | dfss2 2988 | . 2 | |
19 | 16, 17, 18 | 3bitr4g 221 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wal 1282 wcel 1433 cdif 2970 wss 2973 |
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 |
This theorem is referenced by: (None) |
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