Theorem undi 3375

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 3310	. . . 4 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
2	1	orbi2i 757	. . 3 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> ( x e. A \/ ( x e. B /\ x e. C ) ) )
3		ordi 811	. . 3 |- ( ( x e. A \/ ( x e. B /\ x e. C ) ) <-> ( ( x e. A \/ x e. B ) /\ ( x e. A \/ x e. C ) ) )
4		elin 3310	. . . 4 |- ( x e. ( ( A u. B ) i^i ( A u. C ) ) <-> ( x e. ( A u. B ) /\ x e. ( A u. C ) ) )
5		elun 3268	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6		elun 3268	. . . . 5 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
7	5, 6	anbi12i 457	. . . 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 184	. . 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 205	. 2 |- ( ( x e. A \/ x e. ( B i^i C ) ) <-> x e. ( ( A u. B ) i^i ( A u. C ) ) )
10	9	uneqri 3269	1 |- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )

Syntax hints:   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   u. cun 3119   i^i cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127

This theorem is referenced by:  undir  3377  undifdc  6901