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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 4380 | . 2 | |
2 | anass 393 | . . . . 5 | |
3 | 19.42vv 1829 | . . . . . 6 | |
4 | ancom 262 | . . . . . . 7 | |
5 | 4 | 2exbii 1537 | . . . . . 6 |
6 | vex 2604 | . . . . . . . 8 | |
7 | 6 | biantru 296 | . . . . . . 7 |
8 | elvv 4420 | . . . . . . . 8 | |
9 | 8 | anbi2i 444 | . . . . . . 7 |
10 | 7, 9 | bitr3i 184 | . . . . . 6 |
11 | 3, 5, 10 | 3bitr4ri 211 | . . . . 5 |
12 | 2, 11 | bitr3i 184 | . . . 4 |
13 | 12 | 2exbii 1537 | . . 3 |
14 | exrot4 1621 | . . . 4 | |
15 | excom 1594 | . . . . . 6 | |
16 | vex 2604 | . . . . . . . . 9 | |
17 | vex 2604 | . . . . . . . . 9 | |
18 | 16, 17 | opex 3984 | . . . . . . . 8 |
19 | opeq1 3570 | . . . . . . . . 9 | |
20 | 19 | eqeq2d 2092 | . . . . . . . 8 |
21 | 18, 20 | ceqsexv 2638 | . . . . . . 7 |
22 | 21 | exbii 1536 | . . . . . 6 |
23 | 15, 22 | bitri 182 | . . . . 5 |
24 | 23 | 2exbii 1537 | . . . 4 |
25 | 14, 24 | bitr3i 184 | . . 3 |
26 | 13, 25 | bitri 182 | . 2 |
27 | 1, 26 | bitri 182 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 cxp 4361 |
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-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 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-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 |
This theorem is referenced by: ssrelrel 4458 dftpos3 5900 |
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