Theorem iuniin 3831

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 2541	. . . 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 2692	. . . . . 6 |- z e. _V
3		eliin 3826	. . . . . 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 2445	. . . 4 |- ( E. x e. A z e. |^|_ y e. B C <-> E. x e. A A. y e. B z e. C )
6		eliun 3825	. . . . 5 |- ( z e. U_ x e. A C <-> E. x e. A z e. C )
7	6	ralbii 2444	. . . 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 3825	. . 3 |- ( z e. U_ x e. A |^|_ y e. B C <-> E. x e. A z e. |^|_ y e. B C )
10		eliin 3826	. . . 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 3106	1 |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Syntax hints:   <-> wb 104   e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689   C_ wss 3076   U_ciun 3821   |^|_ciin 3822

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-iun 3823  df-iin 3824

This theorem is referenced by: (None)