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Mirrors > Home > ILE Home > Th. List > unissb | Unicode version |
Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) |
Ref | Expression |
---|---|
unissb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 3604 | . . . . . 6 | |
2 | 1 | imbi1i 236 | . . . . 5 |
3 | 19.23v 1804 | . . . . 5 | |
4 | 2, 3 | bitr4i 185 | . . . 4 |
5 | 4 | albii 1399 | . . 3 |
6 | alcom 1407 | . . . 4 | |
7 | 19.21v 1794 | . . . . . 6 | |
8 | impexp 259 | . . . . . . . 8 | |
9 | bi2.04 246 | . . . . . . . 8 | |
10 | 8, 9 | bitri 182 | . . . . . . 7 |
11 | 10 | albii 1399 | . . . . . 6 |
12 | dfss2 2988 | . . . . . . 7 | |
13 | 12 | imbi2i 224 | . . . . . 6 |
14 | 7, 11, 13 | 3bitr4i 210 | . . . . 5 |
15 | 14 | albii 1399 | . . . 4 |
16 | 6, 15 | bitri 182 | . . 3 |
17 | 5, 16 | bitri 182 | . 2 |
18 | dfss2 2988 | . 2 | |
19 | df-ral 2353 | . 2 | |
20 | 17, 18, 19 | 3bitr4i 210 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wal 1282 wex 1421 wcel 1433 wral 2348 wss 2973 cuni 3601 |
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-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-ral 2353 df-v 2603 df-in 2979 df-ss 2986 df-uni 3602 |
This theorem is referenced by: uniss2 3632 ssunieq 3634 sspwuni 3760 pwssb 3761 bm2.5ii 4240 |
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