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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 4556 | . 2 | |
2 | anass 398 | . . . . 5 | |
3 | 19.42vv 1883 | . . . . . 6 | |
4 | ancom 264 | . . . . . . 7 | |
5 | 4 | 2exbii 1585 | . . . . . 6 |
6 | vex 2689 | . . . . . . . 8 | |
7 | 6 | biantru 300 | . . . . . . 7 |
8 | elvv 4601 | . . . . . . . 8 | |
9 | 8 | anbi2i 452 | . . . . . . 7 |
10 | 7, 9 | bitr3i 185 | . . . . . 6 |
11 | 3, 5, 10 | 3bitr4ri 212 | . . . . 5 |
12 | 2, 11 | bitr3i 185 | . . . 4 |
13 | 12 | 2exbii 1585 | . . 3 |
14 | exrot4 1669 | . . . 4 | |
15 | excom 1642 | . . . . . 6 | |
16 | vex 2689 | . . . . . . . . 9 | |
17 | vex 2689 | . . . . . . . . 9 | |
18 | 16, 17 | opex 4151 | . . . . . . . 8 |
19 | opeq1 3705 | . . . . . . . . 9 | |
20 | 19 | eqeq2d 2151 | . . . . . . . 8 |
21 | 18, 20 | ceqsexv 2725 | . . . . . . 7 |
22 | 21 | exbii 1584 | . . . . . 6 |
23 | 15, 22 | bitri 183 | . . . . 5 |
24 | 23 | 2exbii 1585 | . . . 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 1331 wex 1468 wcel 1480 cvv 2686 cop 3530 cxp 4537 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 |
This theorem is referenced by: ssrelrel 4639 dftpos3 6159 |
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