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Theorem xpundi 4061
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 2772 . . . . . 6
21anbi2i 675 . . . . 5
3 andi 815 . . . . 5
42, 3bitri 239 . . . 4
54opabbii 3417 . . 3
6 unopab 3426 . . 3
75, 6eqtr4i 1952 . 2
8 df-xp 4005 . 2
9 df-xp 4005 . . 3
10 df-xp 4005 . . 3
119, 10uneq12i 2783 . 2
127, 8, 113eqtr4i 1959 1
Colors of variables: wff set class
Syntax hints:   wo 359   wa 360   wceq 1425   wcel 1427   cun 2632  copab 3412   cxp 3989
This theorem is referenced by:  xpun 4065  xp2cda 6386  xpcdaen 6389  unctb 6573  alephadd 6577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1342  ax-6 1343  ax-7 1344  ax-gen 1345  ax-8 1429  ax-10 1430  ax-11 1431  ax-12 1432  ax-17 1441  ax-9 1456  ax-4 1462  ax-16 1640  ax-ext 1911
This theorem depends on definitions:  df-bi 175  df-or 361  df-an 362  df-ex 1347  df-sb 1602  df-clab 1917  df-cleq 1922  df-clel 1925  df-v 2337  df-un 2639  df-opab 3413  df-xp 4005
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