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Mirrors > Home > ILE Home > Th. List > elvvv | Unicode version |
Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.) |
Ref | Expression |
---|---|
elvvv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4362 | . 2 | |
2 | anass 381 | . . . . 5 | |
3 | 19.42vv 1788 | . . . . . 6 | |
4 | ancom 253 | . . . . . . 7 | |
5 | 4 | 2exbii 1497 | . . . . . 6 |
6 | vex 2560 | . . . . . . . 8 | |
7 | 6 | biantru 286 | . . . . . . 7 |
8 | elvv 4402 | . . . . . . . 8 | |
9 | 8 | anbi2i 430 | . . . . . . 7 |
10 | 7, 9 | bitr3i 175 | . . . . . 6 |
11 | 3, 5, 10 | 3bitr4ri 202 | . . . . 5 |
12 | 2, 11 | bitr3i 175 | . . . 4 |
13 | 12 | 2exbii 1497 | . . 3 |
14 | exrot4 1581 | . . . 4 | |
15 | excom 1554 | . . . . . 6 | |
16 | vex 2560 | . . . . . . . . 9 | |
17 | vex 2560 | . . . . . . . . 9 | |
18 | 16, 17 | opex 3966 | . . . . . . . 8 |
19 | opeq1 3549 | . . . . . . . . 9 | |
20 | 19 | eqeq2d 2051 | . . . . . . . 8 |
21 | 18, 20 | ceqsexv 2593 | . . . . . . 7 |
22 | 21 | exbii 1496 | . . . . . 6 |
23 | 15, 22 | bitri 173 | . . . . 5 |
24 | 23 | 2exbii 1497 | . . . 4 |
25 | 14, 24 | bitr3i 175 | . . 3 |
26 | 13, 25 | bitri 173 | . 2 |
27 | 1, 26 | bitri 173 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 |
This theorem is referenced by: ssrelrel 4440 dftpos3 5877 |
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