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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 4603 | . 2 | |
2 | anass 399 | . . . . 5 | |
3 | 19.42vv 1891 | . . . . . 6 | |
4 | ancom 264 | . . . . . . 7 | |
5 | 4 | 2exbii 1586 | . . . . . 6 |
6 | vex 2715 | . . . . . . . 8 | |
7 | 6 | biantru 300 | . . . . . . 7 |
8 | elvv 4648 | . . . . . . . 8 | |
9 | 8 | anbi2i 453 | . . . . . . 7 |
10 | 7, 9 | bitr3i 185 | . . . . . 6 |
11 | 3, 5, 10 | 3bitr4ri 212 | . . . . 5 |
12 | 2, 11 | bitr3i 185 | . . . 4 |
13 | 12 | 2exbii 1586 | . . 3 |
14 | exrot4 1671 | . . . 4 | |
15 | excom 1644 | . . . . . 6 | |
16 | vex 2715 | . . . . . . . . 9 | |
17 | vex 2715 | . . . . . . . . 9 | |
18 | 16, 17 | opex 4189 | . . . . . . . 8 |
19 | opeq1 3741 | . . . . . . . . 9 | |
20 | 19 | eqeq2d 2169 | . . . . . . . 8 |
21 | 18, 20 | ceqsexv 2751 | . . . . . . 7 |
22 | 21 | exbii 1585 | . . . . . 6 |
23 | 15, 22 | bitri 183 | . . . . 5 |
24 | 23 | 2exbii 1586 | . . . 4 |
25 | 14, 24 | bitr3i 185 | . . 3 |
26 | 13, 25 | bitri 183 | . 2 |
27 | 1, 26 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cvv 2712 cop 3563 cxp 4584 |
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-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-opab 4026 df-xp 4592 |
This theorem is referenced by: ssrelrel 4686 dftpos3 6209 |
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