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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 3739 | . . . . . 6 | |
2 | 1 | imbi1i 237 | . . . . 5 |
3 | 19.23v 1855 | . . . . 5 | |
4 | 2, 3 | bitr4i 186 | . . . 4 |
5 | 4 | albii 1446 | . . 3 |
6 | alcom 1454 | . . . 4 | |
7 | 19.21v 1845 | . . . . . 6 | |
8 | impexp 261 | . . . . . . . 8 | |
9 | bi2.04 247 | . . . . . . . 8 | |
10 | 8, 9 | bitri 183 | . . . . . . 7 |
11 | 10 | albii 1446 | . . . . . 6 |
12 | dfss2 3086 | . . . . . . 7 | |
13 | 12 | imbi2i 225 | . . . . . 6 |
14 | 7, 11, 13 | 3bitr4i 211 | . . . . 5 |
15 | 14 | albii 1446 | . . . 4 |
16 | 6, 15 | bitri 183 | . . 3 |
17 | 5, 16 | bitri 183 | . 2 |
18 | dfss2 3086 | . 2 | |
19 | df-ral 2421 | . 2 | |
20 | 17, 18, 19 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wex 1468 wcel 1480 wral 2416 wss 3071 cuni 3736 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-in 3077 df-ss 3084 df-uni 3737 |
This theorem is referenced by: uniss2 3767 ssunieq 3769 sspwuni 3897 pwssb 3898 bm2.5ii 4412 sbthlem1 6845 neipsm 12323 neiuni 12330 |
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