HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem xpundi 4101
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.
Assertion
Ref Expression
xpundi

Proof of Theorem xpundi
StepHypRef Expression
1 elun 2779 . . . . . 6
21anbi2i 671 . . . . 5
3 andi 801 . . . . 5
42, 3bitri 238 . . . 4
54opabbii 3454 . . 3
6 unopab 3463 . . 3
75, 6eqtr4i 1941 . 2
8 df-xp 4048 . 2
9 df-xp 4048 . . 3
10 df-xp 4048 . . 3
119, 10uneq12i 2790 . 2
127, 8, 113eqtr4i 1948 1
Colors of variables: wff set class
Syntax hints:   wo 356   wa 357   wceq 1414   wcel 1416   cun 2635  copab 3449   cxp 4032
This theorem is referenced by:  xpun 4106  xp2cda 6610  xpcdaen 6613  alephadd 6830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1331  ax-6 1332  ax-7 1333  ax-gen 1334  ax-8 1418  ax-10 1419  ax-11 1420  ax-12 1421  ax-17 1430  ax-9 1445  ax-4 1451  ax-16 1629  ax-ext 1900
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-ex 1336  df-sb 1591  df-clab 1906  df-cleq 1911  df-clel 1914  df-v 2329  df-un 2642  df-opab 3450  df-xp 4048
Copyright terms: Public domain