Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xpiundir | Unicode version |
Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
xpiundir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2709 | . . . . 5 | |
2 | df-rex 2422 | . . . . . 6 | |
3 | 2 | rexbii 2442 | . . . . 5 |
4 | eliun 3817 | . . . . . . . 8 | |
5 | 4 | anbi1i 453 | . . . . . . 7 |
6 | r19.41v 2587 | . . . . . . 7 | |
7 | 5, 6 | bitr4i 186 | . . . . . 6 |
8 | 7 | exbii 1584 | . . . . 5 |
9 | 1, 3, 8 | 3bitr4ri 212 | . . . 4 |
10 | df-rex 2422 | . . . 4 | |
11 | elxp2 4557 | . . . . 5 | |
12 | 11 | rexbii 2442 | . . . 4 |
13 | 9, 10, 12 | 3bitr4i 211 | . . 3 |
14 | elxp2 4557 | . . 3 | |
15 | eliun 3817 | . . 3 | |
16 | 13, 14, 15 | 3bitr4i 211 | . 2 |
17 | 16 | 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: iunxpconst 4599 resiun2 4839 txbasval 12436 |
Copyright terms: Public domain | W3C validator |