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Theorem xpundi 4087
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 2776 . . . . . 6
21anbi2i 671 . . . . 5
3 andi 801 . . . . 5
42, 3bitri 238 . . . 4
54opabbii 3442 . . 3
6 unopab 3451 . . 3
75, 6eqtr4i 1941 . 2
8 df-xp 4035 . 2
9 df-xp 4035 . . 3
10 df-xp 4035 . . 3
119, 10uneq12i 2787 . 2
127, 8, 113eqtr4i 1948 1
Colors of variables: wff set class
Syntax hints:   wo 356   wa 357   wceq 1414   wcel 1416   cun 2634  copab 3437   cxp 4019
This theorem is referenced by:  xpun 4092  xp2cda 6567  xpcdaen 6570  unctb 6780  alephadd 6784
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 2328  df-un 2641  df-opab 3438  df-xp 4035
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