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Theorem xpundi 4639
 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 4589 . 2
2 df-xp 4589 . . . 4
3 df-xp 4589 . . . 4
42, 3uneq12i 3259 . . 3
5 elun 3248 . . . . . . 7
65anbi2i 453 . . . . . 6
7 andi 808 . . . . . 6
86, 7bitri 183 . . . . 5
98opabbii 4031 . . . 4
10 unopab 4043 . . . 4
119, 10eqtr4i 2181 . . 3
124, 11eqtr4i 2181 . 2
131, 12eqtr4i 2181 1
 Colors of variables: wff set class Syntax hints:   wa 103   wo 698   wceq 1335   wcel 2128   cun 3100  copab 4024   cxp 4581 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-opab 4026  df-xp 4589 This theorem is referenced by:  xpun  4644  djuassen  7135  xpdjuen  7136
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