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.

Step	Hyp	Ref	Expression
1		elin 3686	. . . 4
2	1	orbi2i 519	. . 3
3		ordi 864	. . 3
4		elin 3686	. . . 4
5		elun 3644	. . . . 5
6		elun 3644	. . . . 5
7	5, 6	anbi12i 697	. . . 4
8	4, 7	bitr2i 250	. . 3
9	2, 3, 8	3bitri 271	. 2
10	9	uneqri 3645	1

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