Theorem iuniin 3823

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin	|- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Distinct variable groups:   x, y   y, A   x, B

Allowed substitution hints:   A( x)   B( y)   C( x, y)

Proof of Theorem iuniin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 2538	. . . 4 |- ( E. x e. A A. y e. B z e. C -> A. y e. B E. x e. A z e. C )
2		vex 2689	. . . . . 6 |- z e. _V
3		eliin 3818	. . . . . 6 |- ( z e. _V -> ( z e. |^|_ y e. B C <-> A. y e. B z e. C ) )
4	2, 3	ax-mp 5	. . . . 5 |- ( z e. |^|_ y e. B C <-> A. y e. B z e. C )
5	4	rexbii 2442	. . . 4 |- ( E. x e. A z e. |^|_ y e. B C <-> E. x e. A A. y e. B z e. C )
6		eliun 3817	. . . . 5 |- ( z e. U_ x e. A C <-> E. x e. A z e. C )
7	6	ralbii 2441	. . . 4 |- ( A. y e. B z e. U_ x e. A C <-> A. y e. B E. x e. A z e. C )
8	1, 5, 7	3imtr4i 200	. . 3 |- ( E. x e. A z e. |^|_ y e. B C -> A. y e. B z e. U_ x e. A C )
9		eliun 3817	. . 3 |- ( z e. U_ x e. A |^|_ y e. B C <-> E. x e. A z e. |^|_ y e. B C )
10		eliin 3818	. . . 4 |- ( z e. _V -> ( z e. |^|_ y e. B U_ x e. A C <-> A. y e. B z e. U_ x e. A C ) )
11	2, 10	ax-mp 5	. . 3 |- ( z e. |^|_ y e. B U_ x e. A C <-> A. y e. B z e. U_ x e. A C )
12	8, 9, 11	3imtr4i 200	. 2 |- ( z e. U_ x e. A |^|_ y e. B C -> z e. |^|_ y e. B U_ x e. A C )
13	12	ssriv 3101	1 |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Syntax hints:   <-> wb 104   e. wcel 1480   A.wral 2416   E.wrex 2417   _Vcvv 2686   C_ wss 3071   U_ciun 3813   |^|_ciin 3814

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815  df-iin 3816

This theorem is referenced by: (None)