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Mirrors > Home > ILE Home > Th. List > xpiundi | Unicode version |
Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
xpiundi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom 2595 | . . . 4 | |
2 | eliun 3817 | . . . . . . . 8 | |
3 | 2 | anbi1i 453 | . . . . . . 7 |
4 | 3 | exbii 1584 | . . . . . 6 |
5 | df-rex 2422 | . . . . . 6 | |
6 | df-rex 2422 | . . . . . . . 8 | |
7 | 6 | rexbii 2442 | . . . . . . 7 |
8 | rexcom4 2709 | . . . . . . 7 | |
9 | r19.41v 2587 | . . . . . . . 8 | |
10 | 9 | exbii 1584 | . . . . . . 7 |
11 | 7, 8, 10 | 3bitri 205 | . . . . . 6 |
12 | 4, 5, 11 | 3bitr4i 211 | . . . . 5 |
13 | 12 | rexbii 2442 | . . . 4 |
14 | elxp2 4557 | . . . . 5 | |
15 | 14 | rexbii 2442 | . . . 4 |
16 | 1, 13, 15 | 3bitr4i 211 | . . 3 |
17 | elxp2 4557 | . . 3 | |
18 | eliun 3817 | . . 3 | |
19 | 16, 17, 18 | 3bitr4i 211 | . 2 |
20 | 19 | eqriv 2136 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 wrex 2417 cop 3530 ciun 3813 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-ral 2421 df-rex 2422 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-iun 3815 df-opab 3990 df-xp 4545 |
This theorem is referenced by: xpexgALT 6031 txbasval 12436 |
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