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Theorem undi 3319
 Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undi

Proof of Theorem undi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3254 . . . 4
21orbi2i 751 . . 3
3 ordi 805 . . 3
4 elin 3254 . . . 4
5 elun 3212 . . . . 5
6 elun 3212 . . . . 5
75, 6anbi12i 455 . . . 4
84, 7bitr2i 184 . . 3
92, 3, 83bitri 205 . 2
109uneqri 3213 1
 Colors of variables: wff set class Syntax hints:   wa 103   wo 697   wceq 1331   wcel 1480   cun 3064   cin 3065 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072 This theorem is referenced by:  undir  3321  undifdc  6805
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