Theorem iuniin 3896

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 2583	. . . 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 2740	. . . . . 6 |- z e. _V
3		eliin 3891	. . . . . 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 2484	. . . 4 |- ( E. x e. A z e. |^|_ y e. B C <-> E. x e. A A. y e. B z e. C )
6		eliun 3890	. . . . 5 |- ( z e. U_ x e. A C <-> E. x e. A z e. C )
7	6	ralbii 2483	. . . 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 201	. . 3 |- ( E. x e. A z e. |^|_ y e. B C -> A. y e. B z e. U_ x e. A C )
9		eliun 3890	. . 3 |- ( z e. U_ x e. A |^|_ y e. B C <-> E. x e. A z e. |^|_ y e. B C )
10		eliin 3891	. . . 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 201	. 2 |- ( z e. U_ x e. A |^|_ y e. B C -> z e. |^|_ y e. B U_ x e. A C )
13	12	ssriv 3159	1 |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Syntax hints:   <-> wb 105   e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737   C_ wss 3129   U_ciun 3886   |^|_ciin 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-iun 3888  df-iin 3889

This theorem is referenced by: (None)