Theorem iuniin 3740

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 2478	. . . 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 2622	. . . . . 6 |- z e. _V
3		eliin 3735	. . . . . 6 |- ( z e. _V -> ( z e. |^|_ y e. B C <-> A. y e. B z e. C ) )
4	2, 3	ax-mp 7	. . . . 5 |- ( z e. |^|_ y e. B C <-> A. y e. B z e. C )
5	4	rexbii 2385	. . . 4 |- ( E. x e. A z e. |^|_ y e. B C <-> E. x e. A A. y e. B z e. C )
6		eliun 3734	. . . . 5 |- ( z e. U_ x e. A C <-> E. x e. A z e. C )
7	6	ralbii 2384	. . . 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 199	. . 3 |- ( E. x e. A z e. |^|_ y e. B C -> A. y e. B z e. U_ x e. A C )
9		eliun 3734	. . 3 |- ( z e. U_ x e. A |^|_ y e. B C <-> E. x e. A z e. |^|_ y e. B C )
10		eliin 3735	. . . 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 7	. . 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 199	. 2 |- ( z e. U_ x e. A |^|_ y e. B C -> z e. |^|_ y e. B U_ x e. A C )
13	12	ssriv 3029	1 |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Syntax hints:   <-> wb 103   e. wcel 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619   C_ wss 2999   U_ciun 3730   |^|_ciin 3731

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-iun 3732  df-iin 3733

This theorem is referenced by: (None)