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Mirrors > Home > NFE Home > Th. List > xpundi | Unicode version |
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.) |
Ref | Expression |
---|---|
xpundi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3220 | . . . . . 6 | |
2 | 1 | anbi2i 675 | . . . . 5 |
3 | andi 837 | . . . . 5 | |
4 | 2, 3 | bitri 240 | . . . 4 |
5 | 4 | opabbii 4626 | . . 3 |
6 | unopab 4638 | . . 3 | |
7 | 5, 6 | eqtr4i 2376 | . 2 |
8 | df-xp 4784 | . 2 | |
9 | df-xp 4784 | . . 3 | |
10 | df-xp 4784 | . . 3 | |
11 | 9, 10 | uneq12i 3416 | . 2 |
12 | 7, 8, 11 | 3eqtr4i 2383 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wo 357 wa 358 wceq 1642 wcel 1710 cun 3207 copab 4622 cxp 4770 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-opab 4623 df-xp 4784 |
This theorem is referenced by: xpun 4834 addcdi 6250 |
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