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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 2474 | . . . 4 | |
2 | eliun 3661 | . . . . . . . 8 | |
3 | 2 | anbi1i 431 | . . . . . . 7 |
4 | 3 | exbii 1496 | . . . . . 6 |
5 | df-rex 2312 | . . . . . 6 | |
6 | df-rex 2312 | . . . . . . . 8 | |
7 | 6 | rexbii 2331 | . . . . . . 7 |
8 | rexcom4 2577 | . . . . . . 7 | |
9 | r19.41v 2466 | . . . . . . . 8 | |
10 | 9 | exbii 1496 | . . . . . . 7 |
11 | 7, 8, 10 | 3bitri 195 | . . . . . 6 |
12 | 4, 5, 11 | 3bitr4i 201 | . . . . 5 |
13 | 12 | rexbii 2331 | . . . 4 |
14 | elxp2 4363 | . . . . 5 | |
15 | 14 | rexbii 2331 | . . . 4 |
16 | 1, 13, 15 | 3bitr4i 201 | . . 3 |
17 | elxp2 4363 | . . 3 | |
18 | eliun 3661 | . . 3 | |
19 | 16, 17, 18 | 3bitr4i 201 | . 2 |
20 | 19 | eqriv 2037 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wceq 1243 wex 1381 wcel 1393 wrex 2307 cop 3378 ciun 3657 cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-iun 3659 df-opab 3819 df-xp 4351 |
This theorem is referenced by: xpexgALT 5760 |
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