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Mirrors > Home > ILE Home > Th. List > xpundir | Unicode version |
Description: Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
xpundir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 4604 | . 2 | |
2 | df-xp 4604 | . . . 4 | |
3 | df-xp 4604 | . . . 4 | |
4 | 2, 3 | uneq12i 3269 | . . 3 |
5 | elun 3258 | . . . . . . 7 | |
6 | 5 | anbi1i 454 | . . . . . 6 |
7 | andir 809 | . . . . . 6 | |
8 | 6, 7 | bitri 183 | . . . . 5 |
9 | 8 | opabbii 4043 | . . . 4 |
10 | unopab 4055 | . . . 4 | |
11 | 9, 10 | eqtr4i 2188 | . . 3 |
12 | 4, 11 | eqtr4i 2188 | . 2 |
13 | 1, 12 | eqtr4i 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wo 698 wceq 1342 wcel 2135 cun 3109 copab 4036 cxp 4596 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-opab 4038 df-xp 4604 |
This theorem is referenced by: xpun 4659 resundi 4891 xpfi 6886 xp2dju 7162 hashxp 10728 |
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