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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 3775 | . . . . . 6 | |
2 | 1 | imbi1i 237 | . . . . 5 |
3 | 19.23v 1863 | . . . . 5 | |
4 | 2, 3 | bitr4i 186 | . . . 4 |
5 | 4 | albii 1450 | . . 3 |
6 | alcom 1458 | . . . 4 | |
7 | 19.21v 1853 | . . . . . 6 | |
8 | impexp 261 | . . . . . . . 8 | |
9 | bi2.04 247 | . . . . . . . 8 | |
10 | 8, 9 | bitri 183 | . . . . . . 7 |
11 | 10 | albii 1450 | . . . . . 6 |
12 | dfss2 3117 | . . . . . . 7 | |
13 | 12 | imbi2i 225 | . . . . . 6 |
14 | 7, 11, 13 | 3bitr4i 211 | . . . . 5 |
15 | 14 | albii 1450 | . . . 4 |
16 | 6, 15 | bitri 183 | . . 3 |
17 | 5, 16 | bitri 183 | . 2 |
18 | dfss2 3117 | . 2 | |
19 | df-ral 2440 | . 2 | |
20 | 17, 18, 19 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wex 1472 wcel 2128 wral 2435 wss 3102 cuni 3772 |
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-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-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-in 3108 df-ss 3115 df-uni 3773 |
This theorem is referenced by: uniss2 3803 ssunieq 3805 sspwuni 3933 pwssb 3934 bm2.5ii 4454 sbthlem1 6898 neipsm 12541 neiuni 12548 |
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