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| Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. |
| Ref | Expression |
|---|---|
| xpundi |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elun 2224 |
. . . . . 6
| |
| 2 | 1 | anbi2i 482 |
. . . . 5
|
| 3 | andi 606 |
. . . . 5
| |
| 4 | 2, 3 | bitri 171 |
. . . 4
|
| 5 | 4 | opabbii 2744 |
. . 3
|
| 6 | unopab 2752 |
. . 3
| |
| 7 | 5, 6 | eqtr4i 1540 |
. 2
|
| 8 | df-xp 3264 |
. 2
| |
| 9 | df-xp 3264 |
. . 3
| |
| 10 | df-xp 3264 |
. . 3
| |
| 11 | 9, 10 | uneq12i 2233 |
. 2
|
| 12 | 7, 8, 11 | 3eqtr4i 1547 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: xpun 3311 xp2cda 5078 xpcdaen 5081 alephadd 7792 phtpycolem5 11990 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 997 ax-gen 998 ax-8 999 ax-10 1001 ax-12 1003 ax-17 1006 ax-4 1008 ax-5o 1010 ax-6o 1013 ax-9o 1158 ax-10o 1176 ax-16 1246 ax-11o 1254 ax-ext 1499 |
| This theorem depends on definitions: df-bi 145 df-or 222 df-an 223 df-ex 1016 df-sb 1208 df-clab 1505 df-cleq 1510 df-clel 1513 df-v 1857 df-un 2101 df-opab 2740 df-xp 3264 |