Theorem undi 2252

Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17.

Assertion

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

Proof of Theorem undi

Step	Hyp	Ref	Expression
1		ordi 596	. . . 4 |- ((x e. A \/ (x e. B /\ x e. C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ x e. C)))
2		elin 2207	. . . . 5 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
3	2	orbi2i 255	. . . 4 |- ((x e. A \/ x e. (B i^i C)) <-> (x e. A \/ (x e. B /\ x e. C)))
4		elun 2173	. . . . 5 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
5		elun 2173	. . . . 5 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
6	4, 5	anbi12i 482	. . . 4 |- ((x e. (A u. B) /\ x e. (A u. C)) <-> ((x e. A \/ x e. B) /\ (x e. A \/ x e. C)))
7	1, 3, 6	3bitr4 183	. . 3 |- ((x e. A \/ x e. (B i^i C)) <-> (x e. (A u. B) /\ x e. (A u. C)))
8		elun 2173	. . 3 |- (x e. (A u. (B i^i C)) <-> (x e. A \/ x e. (B i^i C)))
9		elin 2207	. . 3 |- (x e. ((A u. B) i^i (A u. C)) <-> (x e. (A u. B) /\ x e. (A u. C)))
10	7, 8, 9	3bitr4 183	. 2 |- (x e. (A u. (B i^i C)) <-> x e. ((A u. B) i^i (A u. C)))
11	10	eqriv 1474	1 |- (A u. (B i^i C)) = ((A u. B) i^i (A u. C))

Syntax hints:   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958   u. cun 2045   i^i cin 2046

This theorem is referenced by:  undir 2254

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051

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