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Theorem xpundi 4239
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 2893 . . . . . 6
21anbi2i 671 . . . . 5
3 andi 801 . . . . 5
42, 3bitri 238 . . . 4
54opabbii 3589 . . 3
6 unopab 3598 . . 3
75, 6eqtr4i 2053 . 2
8 df-xp 4186 . 2
9 df-xp 4186 . . 3
10 df-xp 4186 . . 3
119, 10uneq12i 2904 . 2
127, 8, 113eqtr4i 2060 1
Colors of variables: wff set class
Syntax hints:   wo 356   wa 357   wceq 1526   wcel 1528   cun 2748  copab 3584   cxp 4170
This theorem is referenced by:  xpun 4244  xp2cda 7016  xpcdaen 7019  alephadd 7420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1443  ax-6 1444  ax-7 1445  ax-gen 1446  ax-8 1530  ax-10 1531  ax-11 1532  ax-12 1533  ax-17 1542  ax-9 1557  ax-4 1563  ax-16 1741  ax-ext 2012
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-ex 1448  df-sb 1703  df-clab 2018  df-cleq 2023  df-clel 2026  df-v 2442  df-un 2755  df-opab 3585  df-xp 4186
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