Theorem indi 3211

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 764	. . . 4 |- ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( ( x e. A /\ x e. B ) \/ ( x e. A /\ x e. C ) ) )
2		elin 3155	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3		elin 3155	. . . . 5 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4	2, 3	orbi12i 713	. . . 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 185	. . 3 |- ( ( x e. A /\ ( x e. B \/ x e. C ) ) <-> ( x e. ( A i^i B ) \/ x e. ( A i^i C ) ) )
6		elun 3113	. . . 4 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
7	6	anbi2i 444	. . 3 |- ( ( x e. A /\ x e. ( B u. C ) ) <-> ( x e. A /\ ( x e. B \/ x e. C ) ) )
8		elun 3113	. . 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 210	. 2 |- ( ( x e. A /\ x e. ( B u. C ) ) <-> x e. ( ( A i^i B ) u. ( A i^i C ) ) )
10	9	ineqri 3159	1 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )

Syntax hints:   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971   i^i cin 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979

This theorem is referenced by:  indir  3213  undisj2  3302  disjssun  3307  difdifdirss  3327  disjpr2  3456  diftpsn3  3527  resundi  4643