Theorem indi 3374

Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem indi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		andi 813	. . . 4 |- ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
2		elin 3310	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3		elin 3310	. . . . 5 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4	2, 3	orbi12i 759	. . . 4 |- ( ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
5	1, 4	bitr4i 186	. . 3 |- ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
6		elun 3268	. . . 4 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
7	6	anbi2i 454	. . 3 |- ( ( x e. A /\ x e. ( B u. C ) ) <-> ( x e. A /\ ( x e. B \/ x e. C ) ) )
8		elun 3268	. . 3 |- ( x e. ( ( A i^i B ) u. ( A i^i C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
9	5, 7, 8	3bitr4i 211	. 2 |- ( ( x e. A /\ x e. ( B u. C ) ) <-> x e. ( ( A i^i B ) u. ( A i^i C ) ) )
10	9	ineqri 3320	1 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i 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:  indir  3376  undisj2  3473  disjssun  3478  difdifdirss  3499  disjpr2  3647  diftpsn3  3721  resundi  4904