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Theorem xpundi 4416
 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 4371 . 2
2 df-xp 4371 . . . 4
3 df-xp 4371 . . . 4
42, 3uneq12i 3125 . . 3
5 elun 3114 . . . . . . 7
65anbi2i 445 . . . . . 6
7 andi 765 . . . . . 6
86, 7bitri 182 . . . . 5
98opabbii 3847 . . . 4
10 unopab 3859 . . . 4
119, 10eqtr4i 2105 . . 3
124, 11eqtr4i 2105 . 2
131, 12eqtr4i 2105 1
 Colors of variables: wff set class Syntax hints:   wa 102   wo 662   wceq 1285   wcel 1434   cun 2972  copab 3840   cxp 4363 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-opab 3842  df-xp 4371 This theorem is referenced by:  xpun  4421
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