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Theorem xpundi 4921
 Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpundi

Proof of Theorem xpundi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4875 . 2
2 df-xp 4875 . . . 4
3 df-xp 4875 . . . 4
42, 3uneq12i 3491 . . 3
5 elun 3480 . . . . . . 7
65anbi2i 676 . . . . . 6
7 andi 838 . . . . . 6
86, 7bitri 241 . . . . 5
98opabbii 4264 . . . 4
10 unopab 4276 . . . 4
119, 10eqtr4i 2458 . . 3
124, 11eqtr4i 2458 . 2
131, 12eqtr4i 2458 1
 Colors of variables: wff set class Syntax hints:   wo 358   wa 359   wceq 1652   wcel 1725   cun 3310  copab 4257   cxp 4867 This theorem is referenced by:  xpun  4926  xp2cda  8049  xpcdaen  8052  alephadd  8441  ustund  18239 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-opab 4259  df-xp 4875
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