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Theorem indi 3328
 Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
indi

Proof of Theorem indi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 andi 808 . . . 4
2 elin 3264 . . . . 5
3 elin 3264 . . . . 5
42, 3orbi12i 754 . . . 4
51, 4bitr4i 186 . . 3
6 elun 3222 . . . 4
76anbi2i 453 . . 3
8 elun 3222 . . 3
95, 7, 83bitr4i 211 . 2
109ineqri 3274 1
 Colors of variables: wff set class Syntax hints:   wa 103   wo 698   wceq 1332   wcel 1481   cun 3074   cin 3075 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082 This theorem is referenced by:  indir  3330  undisj2  3426  disjssun  3431  difdifdirss  3452  disjpr2  3595  diftpsn3  3669  resundi  4840
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