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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 4515 | . 2 | |
2 | df-xp 4515 | . . . 4 | |
3 | df-xp 4515 | . . . 4 | |
4 | 2, 3 | uneq12i 3198 | . . 3 |
5 | elun 3187 | . . . . . . 7 | |
6 | 5 | anbi1i 453 | . . . . . 6 |
7 | andir 793 | . . . . . 6 | |
8 | 6, 7 | bitri 183 | . . . . 5 |
9 | 8 | opabbii 3965 | . . . 4 |
10 | unopab 3977 | . . . 4 | |
11 | 9, 10 | eqtr4i 2141 | . . 3 |
12 | 4, 11 | eqtr4i 2141 | . 2 |
13 | 1, 12 | eqtr4i 2141 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wo 682 wceq 1316 wcel 1465 cun 3039 copab 3958 cxp 4507 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-opab 3960 df-xp 4515 |
This theorem is referenced by: xpun 4570 resundi 4802 xpfi 6786 xp2dju 7039 hashxp 10540 |
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