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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 2225 |
. . . . . 6
| |
| 2 | 1 | anbi2i 483 |
. . . . 5
|
| 3 | andi 607 |
. . . . 5
| |
| 4 | 2, 3 | bitri 171 |
. . . 4
|
| 5 | 4 | opabbii 2745 |
. . 3
|
| 6 | unopab 2753 |
. . 3
| |
| 7 | 5, 6 | eqtr4i 1541 |
. 2
|
| 8 | df-xp 3265 |
. 2
| |
| 9 | df-xp 3265 |
. . 3
| |
| 10 | df-xp 3265 |
. . 3
| |
| 11 | 9, 10 | uneq12i 2234 |
. 2
|
| 12 | 7, 8, 11 | 3eqtr4i 1548 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: xpun 3312 xp2cda 5080 xpcdaen 5083 alephadd 7794 phtpycolem5 12097 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 998 ax-gen 999 ax-8 1000 ax-10 1002 ax-12 1004 ax-17 1007 ax-4 1009 ax-5o 1011 ax-6o 1014 ax-9o 1159 ax-10o 1177 ax-16 1247 ax-11o 1255 ax-ext 1500 |
| This theorem depends on definitions: df-bi 145 df-or 222 df-an 223 df-ex 1017 df-sb 1209 df-clab 1506 df-cleq 1511 df-clel 1514 df-v 1858 df-un 2102 df-opab 2741 df-xp 3265 |