Theorem undi 3398

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	|- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )

Proof of Theorem undi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3333	. . . 4 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	orbi2i 763	. . 3 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> ( x e. A \/ ( x e. B /\ x e. C ) ) )
3		ordi 817	. . 3 |- ( ( x e. A \/ ( x e. B /\ x e. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) )
4		elin 3333	. . . 4 |- ( x e. ( ( A u. B ) i^i ( A u. C ) ) <-> ( x e. ( A u. B ) /\ x e. ( A u. C ) ) )
5		elun 3291	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6		elun 3291	. . . . 5 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
7	5, 6	anbi12i 460	. . . 4 |- ( ( x e. ( A u. B ) /\ x e. ( A u. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) )
8	4, 7	bitr2i 185	. . 3 |- ( ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) <-> x e. ( ( A u. B ) i^i ( A u. C ) ) )
9	2, 3, 8	3bitri 206	. 2 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> x e. ( ( A u. B ) i^i ( A u. C ) ) )
10	9	uneqri 3292	1 |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )

Syntax hints:   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2160   u. cun 3142   i^i cin 3143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150

This theorem is referenced by:  undir  3400  undifdc  6952