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Theorem undi 3720
 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 3649 . . . 4
21orbi2i 521 . . 3
3 ordi 872 . . 3
4 elin 3649 . . . 4
5 elun 3606 . . . . 5
6 elun 3606 . . . . 5
75, 6anbi12i 701 . . . 4
84, 7bitr2i 253 . . 3
92, 3, 83bitri 274 . 2
109uneqri 3608 1
 Colors of variables: wff setvar class Syntax hints:   wo 369   wa 370   wceq 1437   wcel 1872   cun 3434   cin 3435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-in 3443 This theorem is referenced by:  undir  3722  dfif4  3926  dfif5  3927
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