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Theorem undi 3744
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 3686 . . . 4
21orbi2i 519 . . 3
3 ordi 864 . . 3
4 elin 3686 . . . 4
5 elun 3644 . . . . 5
6 elun 3644 . . . . 5
75, 6anbi12i 697 . . . 4
84, 7bitr2i 250 . . 3
92, 3, 83bitri 271 . 2
109uneqri 3645 1
Colors of variables: wff setvar class
Syntax hints:  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  u.cun 3473  i^icin 3474
This theorem is referenced by:  undir  3746  dfif4  3956  dfif5  3957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482
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